The Manin-Peyre conjecture for a certain biprojective cubic threefold
aa r X i v : . [ m a t h . N T ] S e p THE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVETHREEFOLD
VALENTIN BLOMER, J ¨ORG BR ¨UDERN, AND PER SALBERGER
Abstract.
The conjectures of Manin and Peyre are confirmed for a certain threefold. Introduction
The main result.
In a recent memoir [4] we confirmed the predictions of Manin and Peyrefor the distribution of rational points on the cubic fourfold in P defined by(1.1) x y y + x y y + x y y = 0 . Here, we continue our study of this equation, now viewing the polynomial on the left hand side as alinear form in x = ( x , x , x ) and a quadratic form in y = ( y , y , y ). With x and y interpreted ashomogeneous coordinates, the equation (1.1) defines a variety V in P × P . For a Q -rational pointon V there are representatives x , y ∈ Z with ( x ; x ; x ) = ( y ; y ; y ) = 1, both unique up to sign.An anticanonical height function on V is then given by(1.2) H ( x , y ) = max ≤ i,j ≤ | x i | | y j | . Rational points on V ordered with respect to this height accumulate on the subvariety cut out from V by the additional equation x x x y y y = 0. To see this, note that the choices x = (0 , ,
1) and y = ( y , y , − y ) with ( y ; y ) = 1 produce more than B rational points of height at most B on thissubvariety, while on the Zariski-open subset V ◦ of V where x x x y y y = 0 the rational pointsare much sparser. This is a consequence of the following asymptotic formula. Theorem 1.1.
Let N ( B ) denote the number of rational points on V ◦ with height not exceeding B ,and let (1.3) C = Y p (cid:16) − p (cid:17) (cid:16) p + 5 p + 1 p (cid:17) . Then (1.4) N ( B ) = π − CB (log B ) + O (cid:0) B (log B ) − (cid:1) . Hitherto it was only known [2] that N ( B ) ≍ B (log B ) . No effort has been made to optimize theerror term in (1.4). With more work it is possible to show that there is a polynomial P of degreefour and a positive number δ with the property that(1.5) N ( B ) = BP (log B ) + O ( B − δ ) , Mathematics Subject Classification.
Primary 11D45, 11G35, 11M32, 14G05, 14J35.
Key words and phrases.
Manin-Peyre conjecture, cubic threefold, multiple Dirichlet series, universal torsor, crepantresolution.The first author was supported by the Volkswagen Foundation, a Starting Grant of the European Research Counciland a grant from Deutsche Forschungsgemeinschaft. The second author was supported by a grant from DeutscheForschungsgemeinschaft. but in order to keep the paper at reasonable length we have to content ourselves with a detailedproof of (1.4). The reader is referred to Section 1.2 for a brief summary of refinements needed toestablish (1.5).The shape of the asymptotic formula (1.4) is in line with a general conjecture of Manin (see[10]) concerning the distribution of rational points on smooth Fano varieties. However, V has threesingularities located at x i = x j = y i = y j for 1 ≤ i < j ≤
3. A resolution has been obtained in [4,Theorem 4]: Let X ⊂ P × P × P be the triprojective variety defined in trihomogeneous coordinates( x , y , z ) by(1.6) x z + x z + x z = 0 and y z = y z = y z . Then the restriction to X of the projection P × P × P → P × P onto the first two factors is a crepant resolution of V , and one has rk Pic( X ) = 5. In this situation, Batyrev and Tschinkel [1]predict that N ( B ) / ( B (log B ) ) tends to a limit as B → ∞ , and Peyre [19] suggested a formula forthis limit. At the end of this paper, in Sections 7 and 8, we show that our findings in Theorem 1.1agree with Peyre’s formula. Notice in particular that the p -adic factor of the constant C in (1.3) isthe p -adic density of the universal torsor over X , which in our case is a G m -torsor over a P -bundleover a del Pezzo surface of degree six.The Manin-Peyre conjectures for the distribution of rational points on algebraic varieties havereceived considerable attention in recent years. Powerful techniques are available for surfaces (seee.g. the references in [7]). Moreover, the circle method typically produces asymptotic relations thatconfirm the conjectures, but this requires the dimension be large in terms of the degree. If the varietycarries additional structure, further tools can be brought into play. For example, when the varietyis an equivariant compactification of certain linear algebraic groups, Tschinkel and his collaboratorsapplied adelic Fourier analysis to prove Manin’s conjecture in some generality (see e.g. [9, 24]).The variety under consideration is not covered by the cases just described. Definitive results forFano threefolds are very rare besides the remarkable paper of de la Bret`eche [6] on the Segre cubic( x + . . . + x ) = x + . . . x . Le Boudec [5] determined the order of magnitude for the numberof rational points of bounded height on the biprojective threefold x y + x y + x y = 0, and weagree with him that a refinement to an asymptotic formula seems “far out of reach”.Although we are concerned here with just one concrete example, the methods that underpin thethe proof of Theorem 1.1 are by no means restricted to the case at hand. The family of varietiesdefined by x y + x y + · · · + x n y n = 0springs to mind of which we treat the case n = 3 here. Larger n should be within reach for thetechniques described herein, and we intend to return to the theme in a broader setting in due course.We hope that the present example spurs further work on higher dimensional cases of the Manin-Peyreformula.1.2. The methods.
The ideas behind the proof of Theorem 1.1 have some similarity with ourearlier work [4] where the cubic form (1.1) was studied as a fourfold in P . Yet, there are severalfundamental differences. The initial step is the same as in [4]. An elementary argument transfersthe original counting problem to one on the universal torsor. The latter is given by(1.7) u v + u v + u v = 0 , and it is the simple shape of this bilinear equation what makes the variety V accessible to analyticmethods. It is typical that box-like conditions on the original equation transform to regions withhyperbolic spikes on the torsor. In the situation considered here, the anticanonical height function(1.2) involves a product, resulting in very narrow spikes where integral points are difficult to count.This forced us to waive the strategy followed in [4] where the solutions of (1.7) were parametrizedin the obvious way, leading to a hyperbolic lattice point problem that was then approached throughmultiple Dirichlet series. Instead, we use Fourier analysis directly to count points on the torsor. At HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 3 the core of the method, we then require an asymptotic formula for the number N r ( X , Y ) of solutionsto the equation(1.8) r x y + r x y + r x y = 0in x ∈ Z , y ∈ Z within the region B ( X , Y ) = (cid:8) ( x , y ) ∈ R × R | X j < | x j | ≤ X j , Y j < | y j | ≤ Y j , ≤ j ≤ (cid:9) . Here r = ( r , r , r ) ∈ N are given coefficients, X , Y are triples of positive real numbers, andwe need the asymptotic formula uniformly with respect to r . Note that B ( X , Y ) consists of 2 possibly very lopsided boxes. Nonetheless this counting problem is within the competence of thecircle method. The following proposition delivers the desired asymptotics, and we shall save a smallpower of the smallest side of the box. The asymptotic formula involves the singular series(1.9) E r = ∞ X q =1 ϕ ( q )( q ; r )( q ; r )( q ; r ) q and the singular integral(1.10) I r ( X , Y ) = Z R Z B ( X , Y ) e ( α ( r x y + r x y + r x y )) d( x , y ) d α. Proposition 1.2.
Let X , X , X , Y , Y , Y ≥ and r , r , r ∈ N . Then N r ( X , Y ) = E r I r ( X , Y ) + Θ r ( X , Y ) where for any fixed positive value of ε one has Θ r ( X , Y ) ≪ ( r X Y · r X Y · r X Y ) ε max( r X Y , r X Y , r X Y ) min( X , X , X , Y , Y , Y ) / . The implicit constant depends at most on ε . This asymptotic formula is what the circle method predicts, although our proof uses a differentargument in which the key ingredient is a non-trivial bound for Kloosterman sums. The dependenceon r , r , r in the error term Θ r ( X , Y ) can be improved.With this result in hand, one can apply a simple version of the patchwork method developed in[3]. For some small δ >
0, we glue together the contributions from boxes wheremin( X , X , X , Y , Y , Y ) ≥ max( X , X , X , Y , Y , Y ) δ . This keeps us away from the spikes, and then we send δ to 0 at an appropriate speed. In this way,the ideas underpinning the proof [3, Lemma 2.8] deliver the conclusions recorded in Theorem 1.1.The factorization of the leading constant in (1.4) is imported from similar properties of the mainterm in the asymptotics announced in Proposition 1.2. We are fortunate that all local factors canbe computed explicitly.Once Theorem 1.1 is established, we turn to the task of comparing the result with the predictionsmade by Manin and Peyre. In this endeavour, we need to compute some invariants of the crepantresolution X , cf. (1.6). To compute Peyre’s alpha invariant α ( X ), we endow the invertible O X -modules with G m -linearizations compatible with a (non-faithful) G m -action on X induced by theembedding of X in P × P × P . This implies (see Lemma 7.2) that any effective divisor on X is linearly equivalent to a G m -invariant effective divisor. It is then not too hard to show that thepseudo-effective cone in Pic X is spanned by nine G m -invariant prime divisors and to calculate α ( X ).To compute the adelic volume in Peyre’s constant, we proceed as in [4] and relate the volumeforms on the non-singular locus of V to Poincar´e residues of meromorphic forms on P × P withpoles along V . We then see that the Euler product in Theorem 1.1 is the expected one, and alsothat the product of the α -invariant and the archimedean volume agrees with the first factor on theright hand side of (1.4). VALENTIN BLOMER, J ¨ORG BR¨UDERN, AND PER SALBERGER
As we have pointed out before, our arguments can be refined further. Indeed, one may developProposition 1.2 so as to cover the case where some of the variables in (1.8) are fixed, and equippedwith this, one can use the machinery from [3] in full, to cope with the cuspidal portion of the countingmore precisely. This analysis provides error terms that save a power of the largest side of the box.Then, by the main result of [3] one obtains (1.5). Also, one may sum by parts to obtain an analyticcontinuation of the 6-fold Dirichlet series(1.11) X ′ | x | s | x | s | x | s | y | t | y | t | y | t , where the prime indicates summation over ( x , y ) ∈ ( Z \ { } ) satisfying (1.8). If one specialisesto r = r = r = 1 and then restricts to the diagonal s = s = s , t = t = t , this seriesis essentially a minimal parabolic Eisenstein series for SL ( Z ), see [8]. The series (1.11) is a far-reaching generalization of this well-understood Eisenstein series that, apparently, does no longermemorize the group theoretic information carried by its ancestor. Perhaps this is the reason for theconsiderable complexity of our analysis of the threefold defined by (1.1).Details of the arguments outlined in the preceding paragraph will not be worked out in this paper.Armed with this refinement of Proposition 1.2 it would be straightforward but elaborate to do so. Notation . Most of the notation used in this paper is either standard or otherwise explained atthe appropriate stage of the argument. However, traditional notation in the various branches inmathematics on which our work is built resulted in clashes, and the desire for entire consistency inthis respect turned out to be impracticable. The following guide may help the reader to clarify thesymbolism in the work to follow.Throughout, we apply the following convention concerning the letter ε . Whenever ε occurs in astatement, may it be explicitly or implicitly, then it is asserted that the statement is true for anyfixed positive real number in the role of ε . Constants implicit in the use of Landau’s or Vinogradov’swell-known symbols may then depend on the value assigned to ε . Note that this allows us to concludefrom the inequalities A ≪ X ε and B ≪ X ε that AB ≪ X ε , for example.Frequently, we use vector notation x = ( x , . . . , x n ) where the underlying field and the dimension n is usually clear from the context. If the coordinates x j are complex numbers, we write(1.12) | x | = max j | x j | , | x | = n X j =1 | x j | . Further, when x ∈ R n and a ∈ C n we write(1.13) x a = | x | a | x | a . . . | x n | a n . We will often have to integrate over vertical lines in the compex plane. In this context, when c isa real number, the parametrized line ( −∞ , ∞ ) → C , t c + it is denoted by ( c ).The number of divisors of the natural number n is τ ( n ). The M¨obius function is denoted by µ ( n ), and ϕ ( n ) is Euler’s totient. The greatest common divisor of the non-zero integers a , . . . , a n is denoted by ( a ; . . . ; a n ), and their least common multiple is [ a ; . . . ; a n ]. When f : N → C is anarithmetical function and a ∈ N n , we put, by slight abuse of notation, f ( a ) = f ( a ) f ( a ) · · · f ( a n ) . We put Z = Z \ { } . The cardinality of a finite set S is |S| . For a real number θ we put e ( θ ) = exp(2 π i θ ). 2. Auxiliary tools
Smoothing.
To accelerate convergence of certain integrals, we need a smooth approximation tothe characteristic function of the unit interval f : [0 , ∞ ) → [0 , HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 5 convolution argument. For 0 < ∆ < ̺ ∆ : [0 , ∞ ) → [0 , ∞ ) with(2.1) supp( ̺ ∆ ) ⊂ (1 , Z ∞ ̺ ∆ ( x ) d xx = 1such that ̺ ( j )∆ ( x ) ≪ j ∆ − − j holds for all j ∈ N . Its Mellin transform b ̺ ∆ ( s ) = Z ∞ ̺ ∆ ( x ) x s − d x is entire and satisfies(2.2) d j d s j b ̺ ∆ ( s ) ≪ Re s,A (∆ | s | ) − A for all s ∈ C , all j ∈ N (in fact uniformly in j , although we will only apply it for fixed j ) and allintegers A ∈ N , as one confirms by integration by parts and differentiation under the integral sign.For x ∈ [0 , ∞ ) we define(2.3) f ∆ ( x ) = Z ∞ ̺ ∆ ( z ) f (cid:16) xz (cid:17) d zz = Z ∞ x ̺ ∆ ( z ) d zz . Then, by (2.1) and (2.3),(2.4) 0 ≤ f ∆ ( x ) ≤ , f ∆ = 1 on [0 , , supp( f ∆ ) ⊂ [0 , , f ( j )∆ ( x ) ≪ j ∆ − j for x ∈ [0 , ∞ ) and j ∈ N . Further, we have(2.5) b f ∆ ( s ) = b ̺ ∆ ( s ) /s, b f ( s ) = 1 /s. We also note that supp( f ′ ∆ ) ⊂ [1 , f ∆ is indeed a smooth approximation to f , and asin [4, Lemma 24(i)] one shows that(2.6) b f ∆ ( s ) − b f ( s ) ≪ min(∆ , | s | − ) (1 / ≤ Re s ≤ . From (2.5) we now see that(2.7) max (cid:0) b f ∆ ( s ) , b f ( s ) (cid:1) ≪ | s | − (1 / ≤ Re s ≤ . Let D be the differential operator given by ( D f )( x ) = xf ′ ( x ) for differentiable functions f . Thenthe Mellin transforms of D f and f (for, say, Schwartz class functions f ) are related by(2.8) c D f ( s ) = s b f ( s ) . Let X ≥ X ≤ | x | ≤ X . To this end let 1 / ≤ P ≤ X/
10 be another parameter and let v be a non-negative smooth function with v ( x ) = 1 for | x | ∈ [ X − P, X + P ], v ( x ) = 0 for | x | 6∈ [ X − P, X + 2 P ] and k v ( j ) k ∞ ≪ j P − j for all fixed j ∈ N . We call such a function of type ( X, P ). The Mellin transform b v ( s ) of v is entire, and by partial integration one confirms easily thebound(2.9) b v ( s ) ≪ X Re s (1 + | s | P/X ) − in fixed vertical strips. VALENTIN BLOMER, J ¨ORG BR¨UDERN, AND PER SALBERGER
Certain sum transforms.
In this section we consider certain multiple sums with coprimalityconstraints on the variables of summation. Such sums occur in the counting process on the torsor,and we wish to remove the coprimality conditions by M¨obius inversion.Let B denote the set of all ( a , d , z ) ∈ Z × Z × Z that satisfy the coprimality constraints(2.10) ( a z ; a z ; a z ) = ( d i ; d j ) = ( z i ; z j ) = ( d k ; z k ) = 1 (1 ≤ i < j ≤ , ≤ k ≤ , Note that these conditions may also be written in the equivalent form(2.11) ( d i ; d j ) = ( z i ; z j ) = ( d k ; z k ) = 1 (1 ≤ i < j ≤ , ≤ k ≤ , ( a ; a ; a ) = ( a i ; a j ; z k ) = 1 ( { i, j, k } = { , , } ) . The significance of the set B is that it appears naturally in the parametrization of the universaltorsor (see Section 4). Lemma 2.1.
Let G : Z × → C be a function of compact support. Then (2.12) X ( a , d , z ) ∈B G ( a , d , z ) = X b , c , f , g ∈ N ∞ X h =1 µ (( b , c , f , g , h )) X ♯ a , d , z ∈ Z G ( a , d , z ) , in which P ♯ denotes that the sum is restricted to values a , d , z ∈ Z satisfying (2.13) [ g i ; g j ; h ] | a k , [ b i ; b j ; f k ] | d k , [ c i ; c j ; f k ; g k ] | z k ( { i, j, k } = { , , } ) . Proof.
Note that the simultaneous conditions (2.13) are equivalent to the divisibility conditions(2.14) b k | ( d i ; d j ) , c k | ( z i ; z j ) , f k | ( z k ; d k ) , g k | ( a i ; a j ; z k ) , h | ( a ; a ; a ) ( { i, j, k } = { , , } ) . Hence, on applying M¨obius inversion to dissolve all 13 coprimality conditions in (2.11), one obtainsthe desired identity. (cid:3)
In the sum on the right hand side of (2.12) it is often desirable to truncate all sums over b j , c j , f j , g j and h to an interval [1 , T ], say. We wish to control the error in doing so, and for a discussion ofthis matter, some notation is in order. Suppose that the 13 variables b j , c j , f j , g j , h (1 ≤ j ≤
3) arelabelled 1 to 13 in some fixed way, and let S be a non-empty subset of { , , . . . , } . If the label ofsome variable is in S , then we say that the variable belongs to S . If c belongs to S then, by abuseof language, we write c ∈ S , and likewise for other variables.We now claim that the inequality (cid:12)(cid:12)(cid:12) X b , c , f , g ,hx>T if x ∈S µ (( b , c , f , g , h )) X ♯ a , d , z ∈ Z G ( a , d , z ) (cid:12)(cid:12)(cid:12) ≤ X x ∈S x>T X S a , d , z | G ( a , d , z ) | holds, in which P S indicates that the sum is restricted to tuples ( a , d , z ) satisfying the divisibilityconditions (2.14) for those variables that belong to S . Note here that the outer sum consists of |S| independent summations. For a proof of this inequality, we merely have to carry out all summationson the left hand side related to variables that do not belong to S . Reversing the M¨obius inversionformula, we then import one of the conditions (2.11) from each such sum. After this step, we areleft with the summations over variables belonging to S , we apply the triangle inequality and thendrop the imported coprimality constraints to confirm the inequality as claimed above.Equipped with this last inequality, we may indeed truncate all outer sums on the right hand sideof (2.12). The inclusion-exclusion principle then allows us to conclude as follows. Lemma 2.2.
Let T ≥ . In the notation introduced in the preamble to this lemma, one has X ( a , d , z ) ∈B G ( a , d , z ) = X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) X ♯ a , d , z ∈ Z G ( a , d , z ) + O (cid:16)X S X x ∈S x>T X S a , d , z | G ( a , d , z ) | (cid:17) . HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 7
The conditions (2.13) turn out to be significant in the future analysis, and we introduce the3 × α = ( α , α , α ) ∈ N with α j = ( α j , α j , α j ), where whenever { i, j, k } = { , , } ,one takes(2.15) α k = [ g i ; g j ; h ] , α k = [ b i ; b j ; f k ] , α k = [ c i ; c j ; f k ; g k ] . An exponential sum.
In this section we examine an exponential sum of Kloosterman type.When a, b ∈ Z and q ∈ N , the classical Kloosterman sum is defined by S ( a, b ; q ) = q X x =1( x ; q )=1 e (cid:18) ax + b ¯ xq (cid:19) where, here and later, the bar denotes the multiplicative inverse with respect to a modulus thatis always clear from the context; currently this modulus is q . Recall Weil’s classical estimate | S ( a, b ; q ) | ≤ ( a, b, q ) / τ ( q ) q / .For r, x ∈ N , h, h , h ∈ Z we define the sum(2.16) S r,h ( h , h ; x ) = x X ξ,η =1 rξη ≡− h (mod x ) e (cid:18) h ξ + h ηx (cid:19) . We evaluate the sum (2.16) in terms of Kloosterman sums.
Lemma 2.3.
Let r, h, h , h , x be as above. Then one has S r,h ( h , h ; x ) = 0 except when ( r ; x ) | ( h ; h ; h ) , in which case S r,h ( h , h ; x ) equals X d ( r ; x ) | ( x ; h ; h ) d ( r ; x ) S (cid:18) h ( r ; x ) , − h d ( r ; x ) r ( r ; x ) hd ( r ; x ) , xd ( r ; x ) (cid:19) . Proof.
The sum (2.16) is empty unless ( r ; x ) | h , which we assume from now on. We write r ′ = r/ ( r ; x ), h ′ = h/ ( r ; x ) and x ′ = x/ ( r ; x ). Then S r,h ( h , h ; x ) = x X ξ,η =1 ξη ≡− r ′ h ′ (mod x ′ ) e (cid:18) h ξ + h ηx (cid:19) = X d | ( x ′ ; h ′ ) x ′ /d X ξ =1( ξ ; x/d )=1 x X η =1 η ≡− r ′ h ′ d ξ (mod x ′ /d ) e (cid:18) h ξd + h ηx (cid:19) . The sum over η vanishes unless d ( r ; x ) | h , and in the latter case we find that S r,h ( h , h ; x ) = X d ( r ; x ) | ( x ; h ; h ) d ( r ; x ) x/d X ξ =1( ξ ; x ′ /d )=1 e h ξd − h r ′ h ′ d ξx ! . The sum over ξ vanishes unless ( r ; x ) | h , and we obtain the lemma. (cid:3) Lemma 2.4.
Let r, h, h , h , x be as in (2.16) . Then S r,h (0 , x ) = 0 unless ( r ; x ) | h , in which case (2.17) S r,h (0 , x ) = X d ( r ; x ) | ( x ; h ) d ( r ; x ) ϕ (cid:18) x ( r ; x ) d (cid:19) . Further, when h h = 0 , one has the inequalities (2.18) | S r,h (0 , h ; x ) | ≤ τ ( h )( x ; hh ) , (2.19) | S r,h ( h , x ) | ≤ τ ( h )( x ; hh ) , (2.20) | S r,h ( h , h ; x ) | ≤ τ ( x )( r ; x ) x / (cid:16) hh ( r ; x ) ; hh ( r ; x ) ; x (cid:17) / . VALENTIN BLOMER, J ¨ORG BR¨UDERN, AND PER SALBERGER
Proof.
The statements concerning S r,h (0 , x ) are immediate from Lemma 2.3. By symmetry it isenough to prove one of the bounds (2.18) and (2.19), and we show the latter. Since h = 0, thestandard bound for Ramanujan sums | S ( a, q ) | ≤ ( a ; q ) suffices to conclude that | S r,h ( h , x ) | ≤ X d | ( x ( r ; x ) , h ( r ; x ) ) d ( r ; x ) (cid:18) h ( r ; x ) ; xd ( r ; x ) (cid:19) ≤ τ ( h ) (cid:18) x ( r ; x ) ; h ( r ; x ) (cid:19) ( r ; x ) h ( r ; x ) ; x ( r ; x )( x ( r ; x ) ; h ( r ; x ) ) ! = τ ( h )( x ; hh ) . Finally, for h h = 0, Weil’s bound for Kloosterman sums yields | S r,h ( h , h ; x ) | ≤ τ ( x ) X d ( x ; r ) | ( x ; h ; h ) d / ( x ; r ) x / (cid:18) h ; h hd ( r ; x ) ; xd (cid:19) / ≤ τ ( x )( x ; h ; h ) / ( x ; r ) / x / (cid:18) h ; h h ( x ; h ; h ) ; x ( x ; r )( x ; h ; h ) (cid:19) / , and (2.20) follows. (cid:3) Euler products.Lemma 2.5.
Let a ∈ N and X ≥ , / ≤ P ≤ X/ . Further, let v be a function of type ( X, P ) as in Section . Then (2.21) ∞ X n =1 ϕ ( an ) n v ( n ) = ϕ ( a ) ζ (2) Y p | a (cid:18) − p (cid:19) − Z ∞ v ( x ) d xx + O (cid:16) aX / P − log X (cid:17) . Proof.
Comparing Euler products, one easily confirms the formula ∞ X n =1 ϕ ( an ) /ϕ ( a ) n s = ζ ( s − ζ ( s ) Y p | a (cid:18) − p s (cid:19) − in Re s >
2. By Mellin inversion we conclude that X n ≥ ϕ ( an ) n v ( n ) = ϕ ( a ) Z (1) ζ ( s + 1) ζ ( s + 2) Y p | a (cid:18) − p s +2 (cid:19) − b v ( s ) d s π i . We shift the contour to Re s = − /
2. The pole at s = 0 contributes the main term on the righthand side of (2.21). Using (2.9) and Cauchy’s inequality, we bound the remaining integral by ≪ aX / Z ( − / | ζ ( s + 1) | (1 + | s | P/X ) | d s | ≪ aP / Z ( − / | ζ ( s + 1) | (1 + | s | P/X ) | d s | ! / . The standard bound R T | ζ (1 / t ) | d t ≪ T log T [26, Section 2.15] provides the bound ( X/P ) log
X/P for the last integral, which completes the proof. (cid:3)
For r = ( r , r , r ) ∈ N and a prime p let r ( p ) = ( r v p ( r )1 , r v p ( r )2 , r v p ( r )3 ), where v p is the usual p -adic valuation. With the shorthand notation r ′ = r / ( r ; r ), r ′ = r / ( r ; r ) we define(2.22) F r = 1 ζ (2) X abc = r ′ µ ( a ) ab X ( d ; r ′ /b )=1 ( d ; r ′ )( db ; r ) d X fgh = db ( db ; r ( r ; r ) µ ( g ) g Y p | r ′ a ( d ; r ′ (cid:18) −
11 + p (cid:19) . HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 9
The function r
7→ F r / F is multiplicative in r . We define the corresponding Euler factors(2.23) F ( p ) = (cid:18) − p (cid:19) ∞ X δ =0 p δ X ϕ + γ ≤ δγ ≤ ( − γ p γ , F r ( p ) = F ( p ) F r ( p ) F , so that F r = Q p F r ( p ) . Similarly, for the quantity E r defined in (1.9), we define its Euler factors E ( p ) = ∞ X k =0 ϕ ( p k ) p k , E r ( p ) = E ( p ) E r ( p ) E . With this notation, we have the following.
Lemma 2.6. a) Let d ∈ N and r ∈ N . Then E d r = d E r and E r ≪ ( r ; r ; r )( r r r ) ε . b) Let p be a prime and r = ( p α , p β , with ≤ β ≤ α . Then F r ( p ) = E r ( p ) . Proof.
Put α = v p ( r ), β = v p ( r ) and γ = v p ( r ). By symmetry, we may suppose that α ≥ β ≥ γ ,and then find that E r ( p ) = 1 + ∞ X k =1 p k − ( p − p min( k,α )+min( k,β )+min( k,γ ) p k = p γ − α − ( p α ( p + 1)(1 + γ − β + p (1 − γ + β )) − p β +1 ) p + 1 . (2.24)This formula shows on the one hand |E r ( p ) | ≤ p γ (2 + β ), on the other hand we see E r ( p ) = d − E d r ( p )for d = p δ a power of p . Part (a) follows.For (b), we note that F ( p α ,p β , = (cid:18) − p (cid:19) ∞ X d =0 p min( d,β )+min( d,α ) p d X f + g ≤ max(0 ,d − α ) g ≤ ( − g p g X k ≤ min(1 , max(0 ,β − d )) ( − k p k = (cid:18) − p (cid:19) β − X d =0 1 X k =0 ( − k p k + α − X d = β p d − β + ∞ X d = α p d − β − α X f + g ≤ d − αg ≤ ( − g p g = (cid:18) − p (cid:19) (cid:18) βpp + 1 + p − α ( p α − p β ) p − p − α + β (1 + p + p )( p − p + 1) (cid:19) = 1 + β + 1 − βp − p β − α p , which coincides with (2.24) if γ = 0. (cid:3) Our final lemma in this section investigates a multiple sum of multiplicative functions that comesup in the computation of the main term. We recall the definitions (2.15) and (1.3).
Lemma 2.7.
In the range T ≥ one has (2.25) X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) X q ∈ N ϕ ( q ) q Y k =1 ( q ; α k α k ) α k α k α k = C + O ( T ε − ) . Proof.
The product C in (1.3) equals the completed sum on the left of (2.25), with all b j , c j , f j , g j and h running over all natural numbers. We first establish this claim.The completed sum can be written as an Euler product where the Euler p -factor is given (formally)by the same sum, but with all variables of summation running over powers of p . The main observationis that there is no contribution from terms where p | q . Indeed, for squarefree variables b , f , g , h ,the numbers α k and α k are squarefree, and hence, α k α k is cubefree. Then, whenever p | q , wesee that v p (cid:18) ( q ; α k α k ) α k α k α k (cid:19) = v p (cid:18) α k (cid:19) is independent of v p ( h ), and consequently, the contribution from h = 1 and h = p cancel out. Hence,we may introduce the multiplicative factor µ ( q ) in the expression defining the completed sum. Afterthis simplification, a mundane computation shows that the p -th Euler factor of this sum coincideswith that of the product (1.3), as we have claimed.It remains to estimate the error term introduced by completing the sum on the left. To this endwe use Rankin’s trick and bound the characteristic function on x ≥ T by ( x/T ) ξ , for some 0 < ξ < X | b | , | c | , | f | , | g | ,h | µ (( b , c , f , g , h )) | X q ∈ N ϕ ( q ) q Y k =1 ( q ; α k α k ) α k α k α k (cid:16) h ξ + X j =1 ( b ξj + c ξj + f ξj + g ξj ) (cid:17) (2.26)is absolutely convergent. To see this, first note that the rightmost factor in the preceding display isa sum of 13 summands, and it is then sufficient to show absolute convergence with only one of thesesummands present. Irrespective of which summand is present, we are reduced to multiple sum ofmultiplicative terms that we may formally rewrite as an Euler product. As before, its p -th Eulerfactor arises from letting all variables run through powers of p . Again as before, since α k α k iscubefree, it is clear that terms affecting convergence in the Euler p -factor come from q | p . Anothermundane computation then shows that the p -th Euler factor under consideration is of the form1 + O ( p ξ − ). We take ξ = 1 − ε to ensure absolute convergence of the Euler product. This completesthe proof. (cid:3) Mellin inversion formulae.
Our first lemma in this section expresses the Fourier integral(1.10) as a Mellin integral. This features the meromorphic function(2.27) K ( s ) = Γ( s ) cos( πs/ − s − ) (1 − s ) . Lemma 2.8.
Let r ∈ N and X , X , X , Y , Y , Y ≥ . Then, whenever the positive numbers c , c satisfy c + c < , one has I r ( X , Y ) = 64 π Z ( c ) Z ( c ) ( X Y ) − s ( X Y ) − t ( X Y ) s + t r s r t r − s − t K ( s ) K ( t ) K (1 − s − t ) d s d t (2 π i) . In particular, choosing c = c = 1 /
3, we see that(2.28) I r ( X , Y ) ≪ ( X X X Y Y Y ) / ( r r r ) / . Proof.
Let B ( X, Y ) denote the region X ≤ | x | ≤ X , Y ≤ | y | ≤ Y . For α, r ∈ R , one has(2.29) Z B ( X,Y ) e ( αrxy ) d( x, y ) = 2(Si( παrXY ) − παrXY ) + Si(2 παrXY )) παr , where Si( x ) = Z x sin( t ) t d t is the integral sine. By [12, 6.246.1, 8.230.1], the identity(2.30) Z ∞ x/ − x/
2) + Si( x )) x/ x s − d x = 4 Z ( c ) K ( s ) d s π iholds for 0 < c <
1, and hence, for the same c , Mellin inversion yields Z B ( X,Y ) e ( αrxy ) d( x, y ) = 4 Z ( c ) K ( s )(2 πr | α | ) − s ( XY ) − s d s π i . We use this formula twice for the integration over x , y , x , y with contours ( c ), ( c ) such that c , c > c + c <
1. Then we integrate over x , y using (2.29) and finally integrate over α by(2.30). This gives the desired formula. (cid:3) HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 11
The following lemma computes explicitly a certain multiple Mellin integral whose integrand is arational function.
Lemma 2.9.
Fix z = ( z , z ) ∈ C with Re z = Re z = 1 / , and for y = ( y , y , y , y ) ∈ C let F z ( y ) = y y (cid:18) − z − y − y (cid:19) y y (cid:18) − z − y − y (cid:19) (1 − z − z − y − y ) × ( z − y − y ) (cid:18) z + z −
22 + y + y + 2 y + 2 y (cid:19) . Then π i) Z ( ) Z ( ) Z ( ) Z ( ) ( F z ( y )) − d y d y d y d y = 2(1 − z )(1 − z ) z z ( z + z )(1 − z − z ) . Proof.
This can be obtained by straightforward contour shifts. We shift all contours successively tothe far left (the opposite direction would be possible, too). Each time we pick up two poles, and theremaining integral vanishes in the limit. If we first compute the innermost integral over y in thisway and then divide by 2 π i, we arrive at 4(4 y − z + 1)(2 y + z − y + z + z − y + 4 y + z − z )(4 y + 4 y + 2 z + z − × (cid:20) y y y (cid:18) − z − y − y (cid:19) ( z − y − y ) (cid:21) − . We integrate this over y and again divide by 2 π i, then obtaining4 y − z + 1(2 y + z + z − y + 4 y + 2 z + z − y y (cid:0) − z − y − y (cid:1) × z (1 − z )( z + z )(4 y − z + 1)(2 y − z ) . Again, integrating this over y and dividing by 2 π i, one gets the function1 + z (1 − z )( z + z ) y (2 y − z ) · z − z (1 − z − z )(4 y + z − z − − y ) . Finally, Z ( . . . ) d y π i = (1 + z )2( z − − z )( z + z ) z (1 − z − z ) · − z − (1 + z ) z , and the claim follows. (cid:3) The double Mellin integral in the next lemma is related to the archimedean density of our algebraicvariety, cf. Lemma 8.4.
Lemma 2.10.
We have Z ( ) Z ( ) Γ( z )Γ( z )Γ(1 − z − z ) cos( πz ) cos( πz ) cos( π (1 − z − z )2 )(1 − z )(1 − z ) z z ( z + z )(1 − z − z ) d z d z (2 π i) = π π − . Proof.
We call the left hand side I . First we note the Mellin formula Z ∞ (cid:16)Z ∞ y sin tt d t (cid:17) y u − d y = Z ∞ (cid:16)Z ∞ y cos tt d t + sin yy (cid:17) y u − d y = Γ( u ) cos( πu/ u (1 − u )that holds for 0 < Re u < I = Z ∞ (cid:16)Z ∞ y sin tt d t (cid:17) d y = Z ∞ (cid:16)Z ∞ sin ytt d t (cid:17) d yy . We apply the Fubini-Tonelli theorem and see that we may exchange the order of integrations. Thisyields the formula I = Z [1 , ∞ ] T ( t ) t t t d t where T ( t ) = Z ∞ sin yt sin yt sin yt y d y. Let sgn β denote the sign of the real number β . Then, on writing the sin-function in terms ofexponentials, a standard application of the residue theorem shows that T ( t ) = π (cid:0) ( t + t + t ) − ( t + t − t ) sgn( t + t − t ) − ( t − t + t ) sgn( t − t + t ) − ( − t + t + t ) sgn( − t + t + t ) (cid:1) . If t > t > | t − t | ≥
1, a straightforward computation shows (split at | t − t | and t + t ) Z ∞ T ( t ) t t t d t = π t t (cid:18) t , t ) log | t − t | − t t t + t + 4( t + t ) log t + t | t − t | + 8 t t t + t (cid:19) = π ( t + t ) log( t + t ) − | t − t | log | t − t | t t , while for t > t > | t − t | <
1, a slightly simpler computation gives Z ∞ T ( t ) t t t d t = π t t (cid:18) t + t ) log( t + t ) − t + t − t + t + ( t − t ) ) t + t + 8 t t t + t (cid:19) = π (1 − ( t − t ) + 2( t + t ) log( t + t ))8 t t . Let T = { ( t , t ) ∈ (0 , ∞ ) : | t − t | ≥ } , T = { ( t , t ) ∈ (0 , ∞ ) : | t − t | < } . We then have a natural decomposition I = I + I , where I = Z T π ( t + t ) log( t + t ) − | t − t | log | t − t | t t d t , I = Z T π (1 − ( t − t ) + 2( t + t ) log( t + t ))8 t t d t . Obvious substitutions deliver that I = 2 π Z ∞ Z ∞ ( r + 2 t ) log( r + 2 t ) − r log r r + t ) t d t d r = 2 π Z Z ∞ (1 + 2 rt ) log(1 + 2 rt ) − rt log r t r (1 + rt ) d t d r, and a similar computation produces I = 2 π Z Z ∞ − r + 2( r + 2 t ) log( r + 2 t )8( r + t ) t d t d r. The t -integrals in the final expression for I j have an elementary primitive, and a tedious computationyields I = π Z f ( r ) d r where f is defined by − r (1 + r ) f ( r ) =8 r log 2 + r (1 + r )( − r + r + 8 r log 2) + 4 r (1 + r ) log(1 + 1 /r ) log r + 4 r log(4 r ) + 2(1 + r (3 + r + r + 2 r )) log(1 + r ) − r (2 + r ) log(2 + r ) − r (1 + 2 r ) log(1 + 2 r ) . HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 13
Now let Li ( z ) = − Z z log(1 − t ) t d t = ∞ X n =1 z n n be the Dilogarithm, see [16] for basic and more advanced properties of this function. By brute forceone then checks that a primitive of f is given by F , where8 F ( r ) = − r − r + 8 log 2 r − r log 2 + 4 log (cid:18) r (cid:19) + 4 r log (cid:18) r (cid:19) + 4 log r − r log r − r log (cid:18) r (cid:19) − r log r log (cid:18) r (cid:19) − r ) − log(1 + r ) + log(1 + r ) r + 4 log(1 + r ) r − r log(1 + r ) + 2 log r log(1 + r ) − r ) − r ) r + 4 log(1 + 2 r ) − r ) + 8 r log(1 + 2 r ) − r ) log(1 + 2 r ) − ( − − r ) − ( − r ) − ( − r ) − ( − − r ) . In order to evaluate F (1) − F (0), we need to evaluate Li ( −
3) + 2 Li ( − − Li ( − ( −
1) = − π /
12. Moreover, one confirms by differentiation that the function x ( x ) + Li (1 − x ) − ( − /x ) + 2 log(1 − x ) log x − (log x ) is constant, and it takes the value π /
6, as can be seen by substituting x = 1. For x = − ( − /
2) = π − (log 2) [16, (1.16)] to conclude that2 Li ( −
2) + Li ( −
3) = − π − . Altogether, this gives I = π ( F (1) − F (0)) = π ( π − (cid:3) An asymptotic formula
This section is devoted to a proof of Proposition 1.2. By symmetry we can assume that(3.1) r X Y ≤ r X Y ≤ r X Y , X ≤ Y . To begin with, we make the two additional assumptions(3.2) ( r ; r ; r ) = 1 , r X X ≍ r X Y = Z, say.We start our argument by smoothing the summation conditions: let P , P , P , Q , Q , Q satisfy1 / ≤ P j ≤ X j /
10, 1 / ≤ Q j ≤ Y j /
10, and for 1 ≤ j ≤ v j be a function of type ( X j , P j ) and w j be a function of type ( Y j , Q j ), cf. Section 2.1. Let N (1) r ( X , Y ) = X ′ v ( x ) v ( x ) v ( x ) w ( y ) w ( y ) w ( y ) , where the prime indicates summation over ( x , y ) ∈ Z satisfying (1.8). We choose P j = X j Ξ δ , Q j = Y j Ξ δ , Ξ = min( X , X , X , Y , Y , Y )for some 0 < δ < N r ( X , Y ) = N (1) r ( X , Y ) + O (cid:0) X Y Z ε Ξ − δ (cid:1) . Since ( r ; r ; r ) = 1, we may write(3.4) N r ( X , Y ) = X ( r ; r ) | x y v ( x ) w ( y ) M (cid:18) r ( r , r ) , r ( r , r ) , r x y ( r , r ) (cid:19) where M ( r ′ , r ′ , h ) = X h + r ′ x y + r ′ x y =0 v ( x ) v ( x ) w ( y ) w ( y ) . We now manipulate M ( r ′ , r ′ , h ) for h = 0, ( r ′ ; r ′ ) = 1. We have M ( r ′ , r ′ , h ) = X x v ( x ) X r x y ≡− h (mod r ′ | x | ) v ( x ) w (cid:16) − r ′ x y − hr ′ x (cid:17) w ( y )= X x v ( x ) X ξ,η (mod r ′ | x | ) r ′ ξη ≡− h (mod r ′ | x | ) X x ≡ ξ (mod r ′ | x | ) y ≡ η (mod r ′ | x | ) W r ′ ,r ′ ,h ( x , y ; x ) , where W r ′ ,r ′ ,h ( x , y ; x ) = v ( x ) w (cid:16) − r ′ x y − hr ′ x (cid:17) w ( y ) . By the Poisson summation formula we obtain(3.5) M ( r ′ , r ′ , h ) = X x v ( x )( r ′ x ) X h ,h ∈ Z W r ′ ,r ′ ,h (cid:18) h r ′ | x | , h r ′ | x | ; x (cid:19) S r ′ ,h ( h , h ; r ′ | x | ) , where the exponential sum S r ′ ,h ( h , h ; r ′ | x | ) was defined in (2.16), and where W r ′ ,r ′ ,h ( ξ, η ; x ) = Z ∞−∞ Z ∞−∞ W r ′ ,r ′ ,h ( x , y ; x ) e ( − x ξ − y η ) d x d y is the Fourier transform with respect to the first two variables. By partial integration it is easy tosee that W r ′ ,r ′ ,h ( ξ, η ; x ) ≪ A,B X Y (cid:18)(cid:18) P + r ′ Y r ′ Q X (cid:19) | ξ | (cid:19) A (cid:18)(cid:18) Q + r ′ X r ′ Q X (cid:19) | η | (cid:19) B holds for any A, B ≥ x ≍ X . By (2.20), the contribution of the terms h h = 0 to (3.5) istherefore at most(3.6) ≪ Z ε ( r ′ ; h ) / ( r ′ ) / X Y X / (cid:18) P + r ′ Y r ′ Q X (cid:19) (cid:18) Q + r ′ X r ′ Q X (cid:19) . Similarly, by (2.18) and (2.19) the contribution of the terms h h = 0, ( h , h ) = (0 ,
0) is at most(3.7) ≪ Z ε ( r ′ ; h ) r ′ X Y (cid:18) P + 1 Q + r ′ ( X + Y ) r ′ Q X (cid:19) . Moreover, by (2.17), the contribution of the central term equals X ( r ′ ; x ) | h v ( x )( r ′ x ) W r ′ ,r ′ ,h (0 , x ) X d ( r ′ ; x ) | ( r ′ x ; h ) d ( r ′ ; x ) ϕ (cid:18) r ′ x ( r ′ ; x ) d (cid:19) . We substitute this back into (3.4) and sum the error terms (3.6) and (3.7) over x ≍ X and y ≍ Y (recall (3.1)). We continue to write r ′ = r / ( r ; r ) and r ′ = r / ( r ; r ) and see in this way that N (1) r ( X , Y ) equals the expression N (2) r ( X , Y ) = X ( r ; r ) | x y v ( x ) w ( y ) X ( r ′ ; x ) | r x y r r v ( x )( r ′ x ) X d ( r ′ ; x ) | ( r ′ x ; r x y r r ) d ( r ′ ; x ) × ϕ (cid:18) r ′ x ( r ′ ; x ) d (cid:19) Z R Z R v ( x ) w (cid:16) − r x y − r x y r x (cid:17) w ( y ) d x d y , HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 15 up to an error that does not exceed ≪ Z ε ( r ′ ; r ) X Y X Y r ′ (cid:18) P + 1 Q + r ′ ( X + Y ) r ′ Q X (cid:19) + Z ε X Y X Y ( r ′ ; r ) / ( r ′ ) / X / (cid:18) P + r ′ Y r ′ Q X (cid:19) (cid:18) Q + r ′ X r ′ Q X (cid:19) ≪ X Y Z ε (Ξ δ − + Ξ δ − ) . (3.8)In the sum defining N (2) r ( X , Y ), we pull the d -sum outside and introduce a new variable b = ( x ; r ′ ).Then the summation conditions for x , y , x become db ( db ; r ) ( r ; r ) | x y , ( x ; r ′ ) = b, db ( db ; r ′ ) | x . Writing x = x ′ db/ ( d ; r ′ ), the last two conditions are equivalent to ( x ′ d ; r ′ /b ) = 1. Hence we canrewrite the main term as N (2) r ( X , Y ) = X b | r ′ X ( d ; r ′ /b )=1 ( d ; r ′ ) ( r ′ ) d X db ( db ; r ( r ; r ) | x y X ( x ; r ′ /b )=1 ϕ (cid:18) r ′ x ( d ; r ′ ) (cid:19) v ( dbx / ( d ; r ′ )) x × v ( x ) w ( y ) Z R Z R v ( x ) w (cid:16) − r x y − r x y r dbx / ( d, r ′ ) (cid:17) w ( y ) d x d y . By M¨obius inversion this equals X abc = r ′ µ ( a ) a X ( d ; r ′ /b )=1 ( d ; r ′ ) ( r ′ ) d X fgh = db ( r r db ; r µ ( g ) X x ,y X x ϕ (cid:18) r ′ ax ( d ; r ′ ) (cid:19) v ( dbax / ( d ; r ′ )) x × v ( f gx ) w ( hgy ) Z R Z R v ( x ) w (cid:16) − r x y − r f g hx y r dbax / ( d ; r ′ ) (cid:17) w ( y ) d x d y . We execute the x -sum by Lemma 2.5. This introduces an error not exceeding ≪ Z ε X abc = r ′ a X d ( d ; r ′ ) ( r ′ ) d X fgh = db ( r r db ; r X Y f g h X Y r ′ ax ( d ; r ′ ) (cid:18) X ( d ; r ′ ) dba (cid:19) (cid:18) dba ( d ; r ′ ) P + dbaY ( d ; r ′ ) Q X (cid:19) ≪ Z ε ( r ′ ; r ) / r ′ X Y X Y X / (cid:18) P + Y Q X (cid:19) ≪ X Y Z ε Ξ δ − . (3.9)Next we execute the x -sum by Poisson summation and keep only the central term. This introducesan error no larger than ≪ X abc = r ′ a X d ( d ; r ′ ) ( r ′ ) d X fgh = db ( r r db ; r r ′ a ( d ; r ′ ) Y gh X Y (cid:18) X P + r Y X r X Q (cid:19) ≪ Z ε Y X Y r ′ (cid:18) X P + r Y X r X Q (cid:19) ≪ X Y Z ε Ξ δ − ;(3.10)here we applied (3.1). Finally we execute the y -sum by Poisson summation and keep only thecentral term. This introduces an error of(3.11) ≪ Z ε X X Y r ′ (cid:18) Y Q + r Y X r X Q (cid:19) ≪ X Y Z ε Ξ δ − . Hence, up to an error described by (3.9) – (3.11), we can write N (2) r ( X , Y ) in the form N (3) r ( X , Y ) = X abc = r ′ µ ( a ) a X ( d ; r ′ /b )=1 ( d ; r ′ ) ( r ′ ) d X fgh = db ( db ; r ( r ; r ) µ ( g ) ϕ ( r ′ a/ ( d ; r ′ )) ζ (2) × Y p | r ′ a ( d ; r ′ (cid:18) − p (cid:19) − Z R v (cid:18) dbax ( d ; r ′ ) (cid:19) v ( f gx ) w ( hgy ) v ( x ) × w (cid:16) − r x y − r f g hx y r dbax / ( d ; r ′ ) (cid:17) w ( y ) | x | − d( x , x , x , y , y ) . A change of variables yields N (3) r ( X , Y ) = F r J r ( X , Y ) , where F r = r X abc = r ′ µ ( a ) a X ( d ; r ′ /b )=1 ( d ; r ′ ) ( r ′ ) d X fgh = db ( db ; r ( r ; r ) µ ( g ) f g h ϕ ( r ′ a/ ( d ; r ′ )) ζ (2) Y p | r ′ a ( d ; r ′ (cid:18) − p (cid:19) − = 1 ζ (2) X abc = r ′ µ ( a ) ab X ( d ; r ′ /b )=1 ( d ; r ′ )( db ; r ) d X fgh = db ( db ; r ( r ; r ) µ ( g ) g Y p | r ′ a ( d ; r ′ (cid:18) −
11 + p (cid:19) is seen to coincide with the definition (2.22), and where J r ( X , Y ) = 1 r Z R v ( x ) v ( x ) w ( y ) v ( x ) w (cid:16) − r x y − r x y r x (cid:17) w ( y ) | x | − d( x , y , y ) . Note that F r ≪ ( r r r ) ε . Turning to J , we apply Fourier inversion to see that w (cid:16) − r x y − r x y r x (cid:17) = Z R Z R w ( y ) e ( y α ) d y e (cid:18) α r x y + r x y r x (cid:19) d α = r | x | Z R Z R w ( y ) e ( α ( r x y + r x y + r x y )) d y d α. This double integral is not absolutely convergent, but the integral over α is absolutely convergent,and this is all we need to justify the following interchange of integrals: J r ( X , Y ) = Z R Z R v ( x ) w ( y ) v ( x ) w ( y ) v ( x ) w ( y ) e ( α ( r x y + r x y + r x y )) d( x , y ) d α. Finally we remove the smooth weight functions. To this end we observe that the estimate(3.12) Z XX/ Z YY/ e ( αrxy ) d x d y ≪ min (cid:18) XY, r | α | (cid:19) (cf. (2.29)) holds uniformly for r, X, Y ≥ α ∈ R , and one also has Z B ( X,Y ) e ( αrxy ) d( x, y ) − Z R v ( x ) w ( y ) e ( αrxy ) d( x, y ) ≪ min (cid:18) P Y + QX, r | α | (cid:19) , where, as before, B ( X, Y ) denotes the region X ≤ | x | ≤ X , Y ≤ | y | ≤ Y . This shows that J r ( X , Y ) = I r ( X , Y ) + O (cid:0) X Y Z ε Ξ − δ (cid:1) , (3.13)with I r as in (1.10). Collecting the error terms (3.3), (3.8), (3.9), (3.10), (3.11) and (3.13) andchoosing δ = 1 /
6, we have now proved the asymptotic relation(3.14) N r ( X , Y ) = F r I r ( X , Y ) + O (cid:18) X Y ( r X Y ) ε min( X , X , X , Y , Y , Y ) / (cid:19) , HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 17 yet subject to the additional assumptions (3.2).For fixed r and X = X = X = Y = Y = Y = W , it follows easily from Lemma 2.8 that I r ( X , Y ) ≍ r W , while the error term is O r ( W / ε ). Moreover, both N r ( X , Y ) and I r ( X , Y )are symmetric in r , r , r . Thus letting W → ∞ , we conclude that F r is symmetric in r , r , r ,provided ( r ; r ; r ) = 1. By (2.23), also the Euler factors F r ( p ) are symmetric, for all primes p .(This can be checked directly, too, but requires some computation.) By Lemma 2.6b we now inferthat E r = F r if ( r ; r ; r ) = 1, and we have proved Proposition 1.2 under the assumptions (3.2).It remains to remove these extra assumptions. First, should it be the case that(3.15) 10( r X Y + r X Y ) ≤ r X Y , then clearly N r ( X , Y ) = 0. We proceed to show that (3.15) also implies that J r ( X , Y ) = 0. Indeed,formally integrating by parts in the α -integral, we obtain that J r ( X , Y ) = Z R Z R v ( x ) w ( y ) v ( x ) w ( y ) v ( x ) w ( y ) (cid:18) − r x y + r x y r x y (cid:19) n × e ( α ( r x y + r x y + r x y )) d( x , y ) d α for any positive integer n . In particular, we conclude that J r ( X , Y ) = 0 whenever (3.15) holds. Tojustify this formal manipulation, we observe that (by partial integration in any of the x or y -variables)the α -integral is rapidly decaying at ±∞ . Hence we can truncate it (smoothly) with an arbitrarilysmall error, pull it inside and integrate by parts, pull it outside and complete the range of integrationagain with an arbitrarily small error. This argument shows that the proposition holds trivially underthe assumption (3.15), and hence we can drop our initial assumption r X Y ≍ r X Y .By (3.12) we see that the α -integral in the definition of I r ( X , Y ) is absolutely convergent, hencewe can make a change of variables to conclude that I d r ( X , Y ) = d I r ( X , Y )holds for all d ∈ N . Together with Lemma 2.6a we see E d r I d r ( X , Y ) = E r I r ( X , Y )But N d r ( X , Y ) = N r ( X , Y ), whence we may dismiss the assumption that ( r ; r ; r ) = 1. The proofof the proposition is complete.4. The elementary part of the argument
The universal torsor.
We keep the notation introduced in Section 2.2. Let A denote the setof all ( a , d , z ) ∈ Z × N × Z that satisfy the lattice equation(4.1) a d + a d + a d = 0and the coprimality constraints (2.10) or equivalently (2.11). We recall that the four six-tuples( ± x , ± y ) satisfying (1.1) are representatives of the same point on V ◦ . The following result from [2,Section 2] provides a parametrization of the points on V ◦ . Lemma 4.1.
The mapping
A → V ◦ defined by (4.2) x = a z , x = a z , x = a z ,y = d d z , y = d d z , y = d d z is -to- . Upper bounds.
We will use frequently the following lattice point count [14, Lemma 3].
Lemma 4.2.
Let v ∈ Z be primitive and let H i > ≤ i ≤ . Then the number of primitive u ∈ Z that satisfy u v + u v + u v = 0 and that lie in the box | u i | ≤ H i (1 ≤ i ≤ , is O (1 + H H | v | − ) . We introduce the following notation. For X = ( X , X , X ), Y = ( Y , Y , Y ) with X j , Y j ≥ H ≥ r , α , δ , ζ ∈ N let V r ;( α , δ , ζ ) ( X , Y , H ) be the set of 9-tuples ( a , d , z ) ∈ Z satisfying | a j z j | ≤ X j (1 ≤ j ≤ , | d i d j z k | ≤ Y k ( { i, j, k } = { , , } ) , min( | a | , | a | , | a | , | d | , | d | , | d | , | z | , | z | , | z | ) ≤ H, (4.3) r a d + r a d + r a d = 0 , (4.4) α j | a j , δ j | d j , ζ j | z j (1 ≤ j ≤ . (4.5) Lemma 4.3.
Let
H, X j , Y j , r , α , δ , ζ be as in the preceding paragraph, and write Z = | X | + | Y | .Then (4.6) |V r ;( α , δ , ζ ) ( X , Y , H ) | ≪ τ (cid:16) Y j =1 r j α j δ j (cid:17) ( X X X ) / ( Y Y Y ) / ( α α α δ δ δ ) / ζ ζ ζ (log Z ) log H. Proof.
We use some ideas from [2, Section 7]. Changing variables r k r k α k δ k , X k X k α k ζ k , Y k Y k δ i δ j ζ k with { i, j, k } = { , , } , the general version of (4.6) is reduced to the case where α = δ = ζ =(1 , , α , δ , ζ fromthe notation as these are now fixed to (1 , , r ; r ; r ) = 1.We first consider the restricted set ˜ V r ( X , Y , H ) of ( a , d , z ) ∈ V r ( X , Y , H ) satisfying the additionalcondition(4.7) ( r d ; r d ; r d ) = ( r a ; r a ; r a ) = 1 . We cut the a j and d j in dyadic ranges A j < a j ≤ A j and D j < d j ≤ D j . Lemma 4.2 shows thatthe number of ( a , d ) ∈ N satisfying (4.7) and (4.1) in a given dyadic range is at most(4.8) ≪ min( D D D , A A A ) + Q j ( A j D j )max j ( A j D j ) ≪ Y j =1 ( A j D j ) / . Summing this over z = ( z , z , z ) with | z j | ≤ Z j for 1 ≤ j ≤
3, we obtain that for each 6-tuple ofdyadic ranges A j , D j the contribution is(4.9) ≪ min (cid:18) X A , Y D D , Z (cid:19) min (cid:18) X A , Y D D , Z (cid:19) min (cid:18) X A , Y D D , Z (cid:19) Y j =1 ( A j D j ) / . If we define E j = D D D /D j , the above simplifies to(4.10) ≪ Y j =1 A / j E / j min (cid:18) X j A j , Y j E j , Z j (cid:19) . Notice now that as ( ν , ν , ν ) runs over N , the triples ( ν + ν , ν + ν , ν + ν ) take each value in N at most once. Hence we can replace a summation in which the D j = 2 ν j run over powers of 2by a sum in which the E j run over powers of 2. It remains to sum (4.10) over A j and E j which runover powers of 2. For any X, Y, H ≥ X A =2 ν A / E / min (cid:18) XA , YE , H (cid:19) ≪ X / min( Y, HE ) / uniformly in 1 ≤ E ≤ Y , and(4.12) X E =2 ν A / E / min (cid:18) XA , YE (cid:19) ≪ X / Y / uniformly in 1 ≤ A ≤ X .If | a j | ≤ H for some 1 ≤ j ≤
3, then summing (4.10) first over E , E , E using (4.12) and thentrivially over A , A , A , we arrive at | ˜ V r ( X , Y , H ) | ≪ ( X X X ) / ( Y Y Y ) / (log Z ) log H. If | z j | ≤ H for some 1 ≤ j ≤
3, then summing (4.10) first over A , A , A using (4.11) and thentrivially over E , E , E , we arrive again at | ˜ V r ( X , Y , H ) | ≪ ( X X X ) / ( Y Y Y ) / (log Z ) log H. Finally, if | d j | ≤ H for some 1 ≤ j ≤
3, then again we sum (4.10) first over A , A , A using(4.11). Noticing that E i E k H ≪ E j ≪ E i E k , { i, j, k } = { , , } , there are at most (log Z ) log H terms in the sum over E , E , E , and again we obtain(4.13) | ˜ V r ( X , Y , H ) | ≪ ( X X X ) / ( Y Y Y ) / (log Z ) log H. With the above bound for | ˜ V r ( X , Y , H ) | we can easily finish the proof. If ( r a ; r a ; r a ) = a and ( r d ; r d ; r d ) = d , we now apply our bounds with X j ( a ; r r r ) /a in place of X j and Y k ( d ; r r r ) /d in place of Y k . Summing over a and d yields (4.6) in all cases. (cid:3) For
B, H ≥ r , α , δ , ζ ∈ N let V r ;( α , δ , ζ ) ( B, H ) be the set of 9-tuples ( a , d , z ) ∈ Z satisfying(4.14) max ≤ j ≤ ( | a j z j | ) max { i,j,k } = { , , } ( | d i d j z k | ) ≤ B as well as (4.3), (4.4) and (4.5).Summing (4.6) over O (log B ) tuples ( X , Y ) = (4 j , j , j , − j B, − j B, − j B ), we may nowconclude as follows. Lemma 4.4.
For
B, H ≥ and r , α , δ , ζ ∈ N we have |V r ;( α , δ , ζ ) ( B, H ) | ≪ τ (cid:16) Y j =1 r j α j δ j (cid:17) B ( α α α δ δ δ ) / ζ ζ ζ (log B ) log H. A continuous version is given by the following lemma.
Lemma 4.5.
Let
B, H ≥ and let S = S ( B, H ) denote the set of points ( a , d , z ) ∈ [1 , ∞ ) satisfying (4.3) and (4.14) . Then Z S a a a d d d ) / d( a , d , z ) ≪ B (log B ) (log H ) . Proof.
This is a simpler version of the proof of Lemmas 4.3 and 4.4, so we can be brief. We cut thevariables into ranges A j ≤ a j ≤ A j , D j ≤ d j ≤ D j and z j ≤ Z j . Fix 1 ≤ X, Y, ≤ B and considerfirst the contribution of points where a j z j ≤ X for 1 ≤ j ≤ d i d j z k ≤ Y for { i, j, k } = { , , } .Then the integral restricted to this set is ≪ min (cid:18) XA , YD D , Z (cid:19) min (cid:18) XA , YD D , Z (cid:19) min (cid:18) XA , YD D , Z (cid:19) Y j =1 ( A j D j ) / as in (4.9). Arguing as in the proof of Lemma 4.3 with X = X = X = X , Y = Y = Y = Y ,we see that total contribution of all choices A j , D j , Z j is ≪ X Y (log B ) log H , as in (4.13). Finallysumming over O (log B ) tuples ( X , Y ), we complete the proof. (cid:3) The analytic part of the argument
Preliminary transformations.
We begin with some notation. In an effort to establish asufficiently compact presentation we write a typical vector x ∈ C as(5.1) x = ( x , x , x ) = ( x , x , x ; x , x , x ; x , x , x ) . For a typical index we write ℓ = ( i, j ) ∈ { , , } × { , , } . In the notation of the previous sectionswe write ( x , x , x ) = ( a , d , z ). For vectors x = ( x , . . . , x n ), y = ( y , . . . , y n ) we write x · y = ( x y , . . . , x n y n ) . With coordinates on Z given by (5.1), let χ : Z → [0 ,
1] be the characteristic function on theset defined by x x + x x + x x = 0, and let ψ : Z → [0 ,
1] be the characteristic functionon the set of 9-tuples satisfying the coprimality conditions corresponding to (2.11), that is,( x i ; x j ) = ( x i ; x j ) = ( x k ; x k ) = 1 (1 ≤ i < j ≤ , ≤ k ≤ , ( x ; x ; x ) = ( x i ; x j ; x k ) = 1 ( { i, j, k } = { , , } ) . For 0 ≤ ∆ <
1, we then put(5.2) F ∆ ,B ( x ) = Y l =1 Y ≤ i 8. This does not changethe value of N ∆ ( B ), but it is notationally slightly more convenient. Recalling the height condition(1.2), it follows from Lemma 4.1 and (4.2) that N ( B ) = N ( B ), but it is analytically easier to treatthe smooth version N ∆ ( B ) for ∆ > 0. But an asymptotic formula of N ∆ ( B ) with ∆ > N ∆ ( B (1 − ∆)) ≤ N ( B ) ≤ N ∆ ( B )holds. We remove the function ψ , which captures the coprimality conditions, by Lemma 2.2 andestimate the error term by Lemma 4.4 with H = B and r = (1 , , T ≥ N ∆ ( B ) = N ∆ ,T ( B ) + O (cid:16) B (log B ) T ε − / (cid:17) , where N ∆ ,T ( B ) = 132 X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) X x ∈ Z X x ∈ Z X x ∈ Z χ ( α · x ) F ∆ ,B ( α · x ) , (5.5)and α is as in (2.15). The factor T ε − / in the error term of (5.4) comes from observing that forevery subset S in the error term of Lemma 2.2, the corresponding variables x ∈ S occur by Lemma4.4 at least with an exponent 4 / − ε in the denominator.From now on, the analysis will frequently feature multiple Mellin-Barnes integrals over specificvertical lines, and we write R ( n ) for an n -fold iterated such integral; the lines of integration will beclear from the context or otherwise specified in the text. If all n integrations are over the same line( β ), then we write this as R ( n )( β ) .We continue to manipulate N ∆ ,T ( B ). Let ∆ > 0, and recall the definition (5.2). We then useMellin inversion and the notation (1.13) to recast N ∆ ,T ( B ) as132 X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) Z (9)(1) X x , x , x ∈ Z χ ( α · x ) α v x v Y ℓ (cid:16) b f ∆ ( s ℓ ) B s ℓ (cid:17) d s (2 π i) , (5.6) HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 21 where v = v ( s ) = ( v , v , v ) ∈ C × C × C is defined by v = 2( s + s + s ) , v = 2( s + s + s ) , v = 2( s + s + s ) ,v = s + s + s + s + s + s ,v = s + s + s + s + s + s ,v = s + s + s + s + s + s ,v = 2( s + s + s ) + s + s + s ,v = 2( s + s + s ) + s + s + s ,v = 2( s + s + s ) + s + s + s , (5.7)and ℓ runs over { , , } . In view of (2.2) and (2.5), the s -integral in (5.6) is absolutely convergent.At this point it would be possible to evaluate the x -sum directly in terms of Riemann’s zetafunction. This is because χ ( α · x ) is independent of x . However, it is easier to treat x , x , x onequal footing. By partial summation and then unfolding the integral, we have X x , x , x ∈ Z χ ( α · x ) α v x v = 1 α v (cid:16)Y ℓ v ℓ (cid:17) Z [1 , ∞ ) X < | x ℓ |≤ X ℓ χ ( α · x ) X − v − d X = 1 α v (cid:16)Y ℓ v ℓ − − v ℓ (cid:17) Z [1 , ∞ ) X X ℓ < | x ℓ |≤ X ℓ χ ( α · x ) X − v − d X . In the notation of Proposition 1.2 this equals1 α v (cid:16)Y ℓ v ℓ − − v ℓ (cid:17) Z [1 , ∞ ) N α · α ( X , X ) · Y j =1 (cid:18) [ X j ] − h X j i(cid:19) X − v − d X . We would like to evaluate this integral with the aid of Proposition 1.2, and this is successful if wereplace the region [1 , ∞ ) with(5.8) R δ := { x ∈ [1 , ∞ ) : min( x , . . . , x n ) ≥ max( x , . . . , x n ) δ } for 0 < δ < / 10, say. With this in mind, for such δ , we define N ∆ ,T,δ ( B ) = 14 X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) Z (9)(1) α v Y ℓ v ℓ − − v ℓ × Z R δ N α · α ( X , X ) Y j =1 (cid:18) [ X j ] − h X j i(cid:19) X − v − d X Y ℓ b f ∆ ( s ℓ ) B s ℓ d s (2 π i) . (5.9)The next lemma estimates the error that we infer by throwing away the information in the cusps. Lemma 5.1. Uniformly for B ≥ , T ≥ , < ∆ < , < δ < / , one has N ∆ ,T,δ ( B ) = N ∆ ,T ( B ) + O (cid:0) T δB (log B ) (cid:1) . We postpone the proof to the end of this section. We will eventually choose T to be a small powerof log B and δ, ∆ small powers of (log B ) − , see (5.28).5.2. The error term. We are now ready to insert the asymptotic formula from Proposition 1.2,and we also insert the obvious asymptotic formula[ X ] − h X i = X O (1)along with the trivial bound N α · α ( X , X ) ≪ ( X X X X X X ) ε max( X X , X X , X X ) , which follows from a simple divisor argument. This gives(5.10) N α · α ( X , X ) Y j =1 (cid:18) [ X j ] − h X j i(cid:19) = E α · α I α · α ( X , X ) X X X α · α ( X ) , where in the case when X ∈ R δ , one has the estimateΨ α · α ( X ) ≪ X X X Q i =1 Q j =1 ( α ij X ij ) ε max( X X , X X , X X ) min ℓ ( X / ℓ ) ≤ (cid:16) Y i =1 3 Y j =1 α εij (cid:17)(cid:16) Y i =1 3 Y j =1 X / ε − δij (cid:17)(cid:16) Y j =1 X − δ j (cid:17) . (5.11)At this point we see why it is convenient to restrict to the set R δ : the asymptotic formula ofProposition 1.2 provides a power saving with respect to the largest variable because of the inequalitymin ℓ X ℓ ≥ Y ℓ X δ/ ℓ . Inserting the right-hand side of (5.10) into (5.9) yields a corresponding decomposition(5.12) N ∆ ,T,δ ( B ) = N (1)∆ ,T,δ ( B ) + E ∆ ,T,δ ( B ) . In this section we estimate the error term. The bound (5.11) implies the bound Z R δ Ψ α · α ( X ) X v − d X ≪ δ − Y i =1 3 Y j =1 α εij that is valid subject toRe ( v ij ) ≥ − δ 60 (1 ≤ i ≤ , ≤ j ≤ , Re v j ≥ − δ 60 (1 ≤ j ≤ . Let σ = − δ . Shifting all contours to Re s ℓ = σ , we obtain E ∆ ,T,δ ( B ) ≪ B − δ δ X | b | , | c | , | f | , | g | ,h ≤ T (cid:16) Y i =1 3 Y j =1 α + ε + δ ij (cid:17) Z (9)( σ ) Y ℓ | v ℓ b f ∆ ( s ℓ ) || − − v ℓ | | d s | . Also with later applications in mind, we observe that for Re s ℓ ≥ / 100 the bounds (2.2) and (2.5)imply that(5.13) D v ℓ b f ∆ ( s ℓ )1 − − v ℓ ! ≪ D ∆ | s s · · · s | , holds for any differential operator D in the variables s , . . . , s . For now we use this with D = id,getting E ∆ ,T,δ ( B ) ≪ B − δ δ − ∆ − T + ε + δ ) . In particular, we then have(5.14) E ∆ ,T,δ ( B ) ≪ B − δ δ − ∆ − T , uniformly for B, T, δ, ∆ as in Lemma 5.1. HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 23 The main term. We insert the main term in (5.10) into (5.9) getting N (1)∆ ,T,δ ( B ) = 132 X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) Z (9)(1) α v Y ℓ v ℓ b f ∆ ( s ℓ ) B s ℓ − − v ℓ × Z R δ E α · α · I α · α ( X , X ) X X X X − v − d X d s (2 π i) . As a first step we would like to make this independent of δ by replacing R δ (defined in (5.8)) withthe full range [1 , ∞ ) . We write R δ = [1 , ∞ ) \ S δ and obtain a corresponding decomposition(5.15) N (1)∆ ,T,δ ( B ) = N (2)∆ ,T ( B ) − N (2)∆ ,T,δ ( B ) . We anticipate that N (2)∆ ,T,δ ( B ) is small and quantify this in the following lemma. Lemma 5.2. We have N (2)∆ ,T,δ ( B ) ≪ T ∆ − δB (log B ) . Proof. We first consider the s -integral Z (9)(1) α v Y ℓ v ℓ b f ∆ ( s ℓ ) B s ℓ − − v ℓ X − v d s (2 π i) = X n ∈ N Z (9)(1) (2 n · α · X ) − v Y ℓ ( v ℓ b f ∆ ( s ℓ ) B s ℓ ) d s (2 π i) , where of course 2 n is the vector (2 n ℓ ) ℓ ∈{ , , } . By (2.8) and (5.7), the 9-fold inverse Mellin transformof s Q ℓ v ℓ b f ℓ ( s ) is a linear combination of functions of the type x Y ℓ ϕ ℓ ( x ℓ ) , ϕ ℓ = D ν ℓ f ∆ for ν ∈ N with | ν | = 9. Hence by Mellin inversion, the above 9-fold integral is a linear combinationof expressions of the type X n ∈ N ˜ F ∆ ,B (2 n · α · X ) , where ˜ F is defined as in (5.2) but with some of the functions f ∆ replaced with D ν f ∆ . Invokingalso the bounds of Lemma 2.6a and (2.28) along with the trivial bound ( r ; r ; r ) ≤ ( r r r ) / , itsuffices to bound T ε X n ∈ N X | b | , | c | , | f | , | g | ,h ≤ T Z S δ | ˜ F ∆ ,B (2 n · α · X ) | ( X X X X X X ) / d X ≪ T ε ∆ − X n ∈ N Z S δ F , ˜ B ( n ) ( X )( X X X X X X ) / d X with ˜ B ( n ) = B (1 + ∆)2 −| n | . Here we just used the simple observation that each ˜ F is of size O (∆ − )by (2.4) and the above remarks, and f ∆ has support [0 , { X ∈ [1 , ∞ ) | min( X ℓ ) ≤ (2 B ) δ } , so that the desired boundfollows from Lemma 4.5 with H = (2 B ) δ and B = ˜ B ( n ). (cid:3) We now focus on the main term N (1)∆ ,T ( B ) and introduce some notation. Let z , z ∈ C , and let v = ( v , v , v ) as in (5.7). Now define(5.16) w = v + ( z , z , − z − z ) , w = v + ( z , z , − z − z ) , w = v , and put w = ( w , w , w ) ∈ C so that w is a linear function in s and z = ( z , z ). We use (1.9),(1.10) and Lemma 2.8 to write N (2)∆ ,T ( B ) = Z ( ) Z ( ) Z (9)(1) G T ( s , z )Ξ ∆ ( s , z ) B s d s (2 π i) d z d z (2 π i) , where s = X ℓ s ℓ , and where G T ( s , z ) = X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) α w α w α w X q ϕ ( q ) q Y k =1 ( q ; α k α k ) , (5.17) Ξ ∆ ( s , z ) = 2 π K ( z ) K ( z ) K (1 − z − z ) Y ℓ v ℓ b f ∆ ( s ℓ )1 − − v ℓ , (5.18)with K as in (2.27). Here we have quite a bit of flexibility for the s -contours, we only need to makesure that we stay(5.19) to the right of poles of b f ∆ ( s ℓ )(1 − − v ℓ ) − ( w ℓ − − .This is the case, for instance, if Re s ℓ > / ℓ . We make the following affine-linearchange of variables in the s -integral: y = v − (1 − z ) , y = v − (1 − z ) ,y = v − (1 − z ) , y = v − (1 − z ) , y = − s = − X i,j =1 s ij , (5.20)and y , . . . , y are chosen to make the transformation unimodular, e.g.(5.21) y = s , y = s , y = 12 s , y = 12 s . We write A ( y ) = s for the corresponding inverse transformation A , whose Jacobian is 1. This gives(5.22) N (2)∆ ,T ( B ) = Z (11) H T, ∆ ( A ( y ) , z ) B y L ( y ) d( y , z )(2 π i) with the lines of integration defined byRe z j = 1 / , Re y = . . . = Re y = η, Re y = 5 η, Re y = . . . = Re y = 1 / , with H T, ∆ ( A ( y ) , z ) = G T ( A ( y ) , z )Ξ ∆ ( A ( y ) , z )and L ( y ) = Y ℓ ( w ℓ − y y y y (2 y − y − y )(2 y − y − y )( y − y + y )( y − y + y )( y + y − y + y − y ) , and η > η = 10 − ) that we stay to the right of the poles of ( w ℓ − − . Thelines of integration for y , . . . , y are to some extent arbitrary, for instance every line to the right of1 / 18 and to the left of 1 / 12 satisfies Re s ℓ > s ℓ in terms of y j ) andhence is in agreement with the condition (5.19). The fact that the integrand in (5.22) has 9 polarlines with 5 variables y , . . . , y shows that we can obtain at most 9 − HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 25 Lemma 5.3. Let B ≥ , T ≥ , < ∆ < , < δ < / and define (5.23) c T, ∆ = 124 Z (6) H T, ∆ (cid:16) A ( y ) | y = ... = y =0 , z (cid:17) d( y , y , y , y , z , z )(2 π i) , with Re z = Re z = 1 / , Re y = . . . = Re y = 1 / as lines of integration. Then (5.24) N (2)∆ ,T ( B ) = 124 c T, ∆ B (log B ) + O ( T ∆ − B (log B ) ) . The proof is a straightforward, but tedious computation that we postpone to the next section.Combining Lemma 5.3 with Lemma 5.2, (5.15), (5.12) and (5.14), we obtain(5.25) N ∆ ,T,δ ( B ) = 124 c T, ∆ B (log B ) + O (cid:18)(cid:16) T ∆ δ (cid:17) B − δ + ∆ − T B (log B ) (1 + δ log B ) (cid:19) . Computation of the leading constant. In this section we compute the constant c T, ∆ definedin (5.23). First we observe that y = . . . = y = 0 in combination with (5.20) and (5.7) implies(5.26) v = v = (1 − z , − z , z + z ) , v = (1 , , , hence w = w = w = (1 , , 1) by (5.16). Inserting this into (5.17), we conclude from Lemma 2.7with α = 3 / 4, say, that G T (cid:16) A ( y ) | y = ... = y =0 , z (cid:17) = G T = C + O ( T − ) , where C is as in (1.3). Combining (2.27), (5.18), (5.26), and writing s ij in terms of y j by (5.7),(5.20) and (5.21), we find after a short calculation thatΞ ∆ (cid:16) A ( y ) | y = ... = y =0 , z (cid:17) = 2 π K ( z , z ) F ∆ ( y , . . . , y , z , z ) (cid:16) − (cid:17) , where F ∆ ( y , z ) = b f ∆ ( y ) b f ∆ ( y ) b f ∆ (cid:18) − z − y − y (cid:19) b f ∆ (2 y ) b f ∆ (2 y ) b f ∆ (cid:18) − z − y − y (cid:19) × b f ∆ (1 − z − z − y − y ) b f ∆ ( z − y − y ) b f ∆ (cid:18) z + z − 22 + y + y + 2 y + 2 y (cid:19) and K ( z , z ) = Γ( z ) cos (cid:16) πz (cid:17) Γ( z ) cos (cid:16) πz (cid:17) Γ(1 − z − z ) cos (cid:18) π (1 − z − z )2 (cid:19) . We would like to replace F ∆ ( y , z ) with F ( y , z ) and estimate the corresponding error. For Re z j =1 / 3, Re y j = 1 / 15, Stirling’s formula yields the crude bound K ( z , z ) ≪ | z z | − / , and (2.6) – (2.7)deliver the bound F ∆ ( y , z ) − F ( y , z ) ≪ ∆ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − z − y − y (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) − z − y − y (cid:12)(cid:12)(cid:12)(cid:12) | y y y y | (cid:19) − . We now observe that for Re z j = 1 / 3, Re y j = 1 / 15 the integral Z (4) Z (2) | z z | − / (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − z − y − y (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) − z − y − y (cid:12)(cid:12)(cid:12)(cid:12) | y y y y | (cid:19) − | d z | | d y |≪ Z (4) ( | y + y || y + y | ) − | y y y y | − | d y | = (cid:16)Z ( ) Z ( ) | y + y | − | y y | − | d y d y | (cid:17) 26 VALENTIN BLOMER, J ¨ORG BR¨UDERN, AND PER SALBERGER is absolutely convergent, and we conclude that Z (6) Ξ ∆ (cid:16) A ( y ) | y = ... = y =0 , z (cid:17) d( y , y , y , y , z , z )(2 π i) = 16 π Z (6) K ( z , z ) F ( y , . . . , y , z , z ) d( y , y , y , y , z , z )(2 π i) + O (∆ / )where we integrate over Re z j = 1 / 3, Re y j = 1 / 15. We evaluate the y -integral by Lemma 2.9 andthe z -integral by Lemma 2.10. The above discussion now delivers c T, ∆ = 124 · π (cid:16) C + O ( T − / ) (cid:17) (cid:16) π π − O (∆ / ) (cid:17) = π − C + O (cid:16) ∆ / + T − / (cid:17) . (5.27)5.5. The endgame. We are ready to complete the proof of Theorem 1.1. Collecting (5.4), Lemma5.1, (5.25) and (5.27), we find that N ∆ ( B ) = N ∆ ,T,δ ( B ) + O (cid:18) B (log B ) (cid:16) T / + T δ (cid:17)(cid:19) = π − · CB (log B ) + O (cid:18) B (log B ) (cid:16) T / + ∆ / + T ∆ − (cid:16) δ + 1log B (cid:17)(cid:17) + (cid:16) T ∆ δ (cid:17) B − δ (cid:19) . By (5.3), the passage from N ∆ ( B ) to N ( B ) introduces an error ∆ B (log B ) that is already presentin the above asymptotic formula, hence the same formula holds for N ( B ). We now choose(5.28) δ = 1(log B ) / , ∆ = 1(log B ) / , T = (log B ) / . to complete the proof of Theorem 1.1, but it remains to provide proofs for Lemmas 5.1 and 5.3.5.6. Proof of Lemma 5.1. In order to estimate the difference between N ∆ ,T,δ ( B ) and N ∆ ,T ( B ),we reverse the steps between (5.5) and (5.9) and eventually use Lemma 4.4. Starting from (5.9), wehave N ∆ ,T,δ ( B ) = 132 X | b | , | c | , | f | , | g | ,h ≤ T µ (( b , c , f , g , h )) Z (9)(1) α v Y ℓ (cid:16) v ℓ − − v ℓ b f ∆ ( s ℓ ) B s ℓ (cid:17) × (cid:16)Z R δ X X ℓ < | x ℓ |≤ X ℓ ℓ ∈{ , , } χ ( α · x ) X − v − d X (cid:17) d s (2 π i) with α as in (2.15). For a vector σ ∈ { , } we define X σ = (2 σ ℓ X ℓ ) ℓ ∈{ , , } and R δ ( σ ) = { X ∈ [1 , ∞ ) | min(2 σ ℓ X ℓ ) ≥ max(2 σ ℓ X ℓ ) δ } . By a change of variables, the integral over R δ equals X σ ∈{ , } ( − | σ | Z R δ ( σ ) X < | x ℓ |≤ X ℓ χ ( α · x ) X − v − σ d X , and by partial summation this equals (cid:16)Y ℓ v ℓ (cid:17) X σ ∈{ , } ( − | σ | − P ℓ σ ℓ v ℓ X ( σ ,δ )+ x ∈ Z χ ( α · x ) x v , HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 27 where P ( σ ,δ )+ denotes a summation over x satisfyingmin ℓ (2 σ ℓ | x ℓ | ) ≥ max ℓ (2 σ ℓ | x ℓ | ) δ , and correspondingly we write P ( σ ,δ ) − for a summation with the opposite condition(5.29) min ℓ (2 σ ℓ | x ℓ | ) < max ℓ (2 σ ℓ | x ℓ | ) δ . In this this notation, one has N ∆ ,T ( B ) − N ∆ ,T,δ ( B ) ≪ M ∆ ,T,δ ( B ) where M ∆ ,T,δ ( B ) = X | b | , | c | , | f | , | g | ,h ≤ T X σ ∈{ , } (cid:12)(cid:12)(cid:12)Z (9)(1) − P ℓ σ ℓ v ℓ X ( σ ,δ ) − x ∈ Z χ ( α · x ) α v x v Y ℓ b f ∆ ( s ℓ ) B s ℓ − − v ℓ d s (2 π i) (cid:12)(cid:12)(cid:12) . We write the factor (1 − − v ℓ ) − as a geometric series and apply Mellin inversion to recast theintegral as X k ∈ N X ( σ ,δ ) − x ∈ Z χ ( α · x ) F ∆ ,B (cid:0) α · (2 k ℓ + σ ℓ x ℓ ) ℓ (cid:1) , so that M ∆ ,T,δ ( B ) = X | b | , | c | , | f | , | g | ,h ≤ T X x ∈ Z χ ( α · x ) G α ( x )with G α ( x ) = X σ ∈{ , } X k ∈ N F ∆ ,B ( α · (2 k ℓ + σ ℓ x ℓ ) ℓ )Ψ σ ,δ ( x ) , in which Ψ σ ,δ is the characteristic function of the set defined by (5.29). In particular,supp G α ⊆ { x ∈ R | min( | x ℓ | ) ≤ (2 B (1 + ∆)) δ } , G α ( x ) ≪ X k ∈ N F ,B (1+∆) (cid:0) α · (2 k ℓ x ℓ ) ℓ (cid:1) . By Lemma 4.4 with (2 B (1 + ∆)) δ in place of H and B (1 + ∆) in place of B we conclude that M ∆ ,T,δ ( B ) ≪ X | b | , | c | , | f | , | g | ,h ≤ T X k ∈ N − ( − ε ) P ℓ k ℓ B (log B ) log( B δ ) ≪ δT B (log B ) , as desired. 6. Proof of Lemma 5.3 We start with the evaluation of N (2)∆ ,T ( B ), defined in (5.22), and prove (5.24). We perform variouscontour shifts with the variables y , . . . , y . The variables y , . . . , y , z , z will be kept fixed. Wewill always stay in the region | Re y | , . . . , | Re y | ≤ η with η = 10 − as before, and we rememberour choice Re z = Re z = 1 / 3. In this region we have Re w ℓ ≥ − η , as one can check from (5.7),(5.16), (5.20) and (5.21), and we derive now rather crude, but convenient bounds for the function H T, ∆ ( s , z ) and its derivatives appearing in the integrand of (5.22) (the derivatives are needed forresidue computations). Let D j denote a differential operator of degree j in s , . . . , s . Then forRe w ℓ ≥ − η we obtain by the most trivial estimates(6.1) D j G T ( s , z ) ≪ j X | b | , | c | , | f | , | g | ,h ≤ T Y k =1 α − Re w k + ε k α − Re w k + ε k ≪ T η + ε ≪ T . Similarly, for Re s ℓ > / 100 we conclude from (5.13) that(6.2) D j Ξ ∆ ( s , z ) ≪ j ∆ − | s s · · · s z z | − . We now shift successively the y , . . . , y contours to Re y j = − jη , 1 ≤ j ≤ 4, thereby pickingup a simple pole at 0 and a remaining integral. This leaves us altogether with 16 terms, someof which are identical by symmetry. We denote by V ⊆ { y , y , y , y } the set of variables that have not been integrated out and distinguish several cases. For notational simplicity we write˜ H ( y , . . . , y ) = H T, ∆ ( A ( y ) , z ) . Case I: V = ∅ . The term consisting only of residues equals Z (5 η ) ˜ H (0 , , , , y ) B y y d y π i = 14 ˜ H ( ) (log B ) 4! + O ((log B ) T ∆ − )by shifting the contour to Re y = − η , say, and spelling out the leading term of the residue, whileestimating the lower order terms and the remaining integral trivially (and crudely) by (6.1) and(6.2).6.2. Case II: V = { y } . Here we have Z ( − η ) Z (5 η ) ˜ H ( y , , , , y ) B y y y ( y − y )(2 y − y )( y + y ) d y d y (2 π i) . Shifting the line Re y = 5 η to Re y = − η/ 3, we pick up a pole at y = − y and y = 0, the latter ofwhich as well as the remaining integral we estimate trivially. Hence the previous expression equals Z ( − η ) ˜ H ( y , , , , B − y y d y π i + O ( T ∆ − log B ) = − 112 ˜ H ( ) (log B ) 4! + O ( T ∆ − (log B ) )which we realize after shifting the line of integration to Re y = η and spelling out only the leadingterm of the residue at y = 0. The same evaluation holds for V = { y } , V = { y } , V = { y } .6.3. Case IIIa: V = { y , y } . In Z ( − η ) Z ( − η ) Z (5 η ) ˜ H ( y , y , , , y ) B y y y y ( y + y − y )( y − y + y )(2 y − y )(2 y − y ) d y d y d y (2 π i) we shift the line Re y = 5 η to Re y = − η/ O ( T ∆ − )coming from the simple pole at y = 0, we pick up the residue at y = y − y , which equals Z ( − η ) Z ( − η ) ˜ H ( y , y , , , y − y ) B y − y y y ( y − y ) (2 y − y )( y − y ) d y d y (2 π i) = O ( T ∆ − log B ) , as we see from shifting Re y to − / η and estimating trivially the contribution of the double poleat y = y . The same bound holds by symmetry for V = { y , y } .6.4. Case IIIb: V = { y , y } . In Z ( − η ) Z ( − η ) Z (5 η ) ˜ H ( y , , y , , y ) B y y y y ( y + y )( y − y − y )(2 y − y − y )( y + y ) d y d y d y (2 π i) we shift the line Re y = 5 η to Re y = − η . Up to an error of O ( T ∆ − ) for the simple pole at y = 0 we get contributions from two poles at y = − y and y = − y . The former yields Z ( − η ) Z ( − η ) ˜ H ( y , , y , , − y ) B − y y y ( y − y )(2 y + y )(3 y + y ) d y d y (2 π i) = O ( T ∆ − log B ) , as one finds after shifting the line Re y = − η to Re y = η/ y = 0. The latter yields Z ( − η ) Z ( − η ) ˜ H ( y , , y , , − y ) B − y y y ( y − y )( y + 2 y )( y + 3 y ) d y d y (2 π i) . We shift the line Re y = − η to Re y = η/ 4. Up to an error of O ( T ∆ − log B ), we get acontribution of the pole at y = y , and its residue is − Z ( − η ) ˜ H ( y , , y , , − y ) B − y y d y π i = 124 ˜ H ( ) (log B ) 4! + O ( T ∆ − (log B ) ) . HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 29 Case IIIc: V = { y , y } . In Z ( − η ) Z ( − η ) Z (5 η ) ˜ H (0 , y , y , , y ) B y y y (2 y − y )( y − y )(2 y − y )( y + y − y )( y + y ) d y d y d y (2 π i) we shift the line Re y = 5 η to Re y = − η/ 2. We pick up one pole at y = − y that contributes Z ( − η ) Z ( − η ) ˜ H (0 , y , y , , − y ) B − y y y (2 y − y )( y + y )( y + 2 y ) d y d y (2 π i) . We shift the line Re y = − η to Re y = η/ 2. The pole at y = 0 contributes O ( T ∆ − log B ),and the residue at y = y / − Z ( − η ) ˜ H (0 , y , y / , , − y / B − y / y d y π i = 172 ˜ H (0) (log B ) 4! + O ( T ∆ − (log B ) ) , as is readily confirmed by shifting the line of integration to the far right. The same evaluation holdsfor V = { y , y } .6.6. Case IIId: V = { y , y } . We consider Z ( − η ) Z ( − η ) Z (5 η ) ˜ H (0 , y , , y , y ) B y y y y ( y − y )( y − y )(2 y − y − y )( y + y + y ) d y d y d y (2 π i) . We begin by moving Re y = 5 η to Re y = − η . Observing the pole at y = − y − y , we then seethat this integral equals Z ( − η ) Z ( − η ) ˜ H (0 , y , , y , − y − y ) B − y − y y y ( y + y ) (2 y + y )( y + 2 y ) d y d y (2 π i) + O ( T ∆ − ) . Next, by shifting Re y = − η to Re y = 5 η , we pick up three residues and a remaining integral ofsize O ( T ∆ − B − η ). The pole at y = − y contributes O ( T ∆ − log B ), the pole at y = 0 gives − Z ( − η ) ˜ H (0 , , , y , − y ) B − y y d y π i = 112 ˜ H ( ) (log B ) 4! + O ( T ∆ − (log B ) ) , and the pole at y = − y / Z ( − η ) ˜ H (0 , − y / , , y − y / B − y / y d y π i = − 136 ˜ H ( ) (log B ) 4! + O ( T ∆ − (log B ) ) . Case IVa: V = { y , y , y } . We wish to evaluate Z (4) ˜ H (0 , y , y , y , y ) B y y y y ( y + y − y )( y + y − y + y )( y − y )( y − y )( y + y − y ) d( y , y , y , y )(2 π i) , with integrations over Re y = − η , Re y = − η , Re y = − η , Re y = 5 η . First, we shiftRe y = 5 η to Re y = − η/ 2. The remaining integral is O ( T ∆ − B − η/ ) and we pick up a pole at y = − y + y − y with residue Z (3) ˜ H (0 , y , y , y , − y + y − y ) B − y + y − y y y y ( y − y + 2 y )(2 y − y + y )(2 y − y + 2 y )(3 y − y + 3 y ) d( y , y , y )(2 π i) ;here the integrations are over the same lines as before. Next, we shift Re y = − η to Re y = − η .The remaining integral then is O ( T ∆ − B − η ), and we pick up a pole at y = (2 y + y ) / − Z ( − η ) Z ( − η ) ˜ H (0 , y , y + y / , y , − y / B − y / y y (2 y + y )( y + 2 y )(3 y + 2 y ) d y d y (2 π i) = O ( T ∆ − log B ) , as is readily seen after shifting Re y = − η to Re y = η . The same bound holds, by symmetry, for V = { y , y , y } . Case IVb: V = { y , y , y } . In this case we consider Z (4) ˜ H ( y , , y , y , y ) B y y y y ( y − y + y )( y − y − y + y )( y + y )( y − y )( y + y − y ) d( y , y , y , y )(2 π i) , with integrations over the linesRe y = − η , Re y = − η , Re y = − η , Re y = 5 η . As in theprevious case, we shift Re y = 5 η to Re y = − η/ 2. The remaining integral is O ( T ∆ − B − η/ ),the pole at y = y + y − y contributes O ( T ∆ − ), and we are left with the pole at y = − y .The latter has residue Z (3) ˜ H ( y , , y , y , − y ) B − y y y y ( y − y + y )(2 y + y − y )(2 y + y )(3 y + y ) d( y , y , y )(2 π i) , with lines of integrations as before. We shift Re y = − η to Re y = η/ 2. The remaining integral is O ( T ∆ − B − η/ ), the pole at y = 0 contributes O ( T ∆ − ), and we only need to consider thepole at y = ( y − y ) / − Z ( − η ) Z ( − η ) ˜ H (( y − y ) / , , y , y , ( y − y ) / B ( y − y ) / ( y − y ) ( y − y )( y − y ) d y d y (2 π i) . Now we shift Re y = − η to Re y = − η . The remaining integral is O ( T ∆ − B − η ), and thepole at y = y contributes O ( T ∆ − log B ). The same bound holds in the case V = { y , y , y } .6.9. Case V: V = { y , y , y , y } . Finally, in the case where none of the variables has been inte-grated out, we shift Re y = 5 η to − η/ 2; the remaining integral is O ( T ∆ − B − η/ ), and we pickup a pole with residue Z (4) ˜ H ( y , y , y , y , y − y + y − y ) B y − y + y − y y y y y ( y − y + 2 y − y )(2 y − y + y − y ) × y − y + 2 y − y )( y − y + y − y ) d( y , y , y , y )(2 π i) . Here, all lines of integration are given by Re y j = − jη . We shift Re y to − / η . The remainingintegral is O ( T ∆ − B − η/ ), and we pick up a simple pole at y = y − ( y − y ) / − Z (3) ˜ H ( y − ( y − y ) / , y , y , y , ( y − y ) / B ( y − y ) / y y y ( y − y )( − y + y − y )( − y + y − y )( − y + y − y ) d( y , y , y )(2 π i) , with lines of integration still given by Re y j = − jη . Next we shift the line for y to Re y − η . Theremaining integral is O ( T ∆ − B − η ), and the pole at y = y contributes O ( T ∆ − ).Summarizing all previous calculations, we obtain (5.24) from Cases I, II (with multiplicity 4),IIIb, IIIc (with multiplicity 2) and IIId, since14 − 412 + 124 + 272 + 112 − 136 = 124 . The geometry of the crepant resolution Let X ⊂ P × P × P be the smooth triprojective variety described in (1.6) with trihomogeneouscoordinates ( x , y , z ) = ( x , x , x ; y , y , y ; z , z , z ). The aim of this chapter is to compute Peyre’salpha invariant of X . We will not specify the base field as the results in this chapter are purelyalgebraic and independent of the base field.Along with X we consider the non-singular biprojective surface Y ⊂ P × P defined in biho-mogeneous coordinates ( y , z ) by y z = y z = y z , and the subvariety Z ⊂ P × P defined inbihomogeneous coordinates ( x , z ) by x z + x z + x z = 0. We also recall that V ⊂ P × P is the HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 31 singular biprojective cubic threefold with bihomogeneous coordinates ( x , y ) = ( x , x , x ; y , y , y )as in (1.1). There are natural projections p : X → Y, g : X → Z, f : X → V defined by p : ( x , y , z ) ( y , z ), g : ( x , y , z ) ( x , z ), f : ( x , y , z ) ( x , y ). We will frequentlyuse these maps and its corresponding induced functorial maps. We will also use the G m -action on P × P × P defined by γ ( x , y , z ) = ( γ x , γ x , γ x ; γ y , γ y , γ y ; γ − z , γ − z , γ − z )for γ =( γ , γ , γ ) ∈ G m , and its restriction to G m -actions on X , P × P and Z , the latter two givenby(7.1) γ ( x , z ) = ( γ x , γ x , γ x ; γ − z , γ − z , γ − z ) . The morphism g is then G m -equivariant and the base extension of the morphism h : Y → P ,( y , z ) z along the second projection pr : Z → P . As h is the blow-up of P at the three pointswhere two of the z -coordinates vanish, we thus obtain that g is the blow-up at the union of the threedisjoint lines l i on Z defined by(7.2) l i : x i = z j = z k = 0for { i, j, k } = { , , } .7.1. The pseudoeffective cone. Nine integral subsurfaces of X will be important for the compu-tation of α ( X ): if 1 ≤ i ≤ { j, k } = { , , } \ { i } , then D i ⊂ P × P × P is defined by the equations x i = y i = z j = z k = 0, E i ⊂ P × P × P is defined by the equations x j z j + x k z k and y j = y k = z i = 0, F i ⊂ P × P × P is defined by the equations x i = x j y k + x k y j = x j z j + x k z k = 0 and y j z j − y i z i = y k z k − y i z i = 0.Here D i is isomorphic to P × P while E i is a P -bundle over the line in P × P with coordinates( y , z ) defined by y j = y k = z i = 0. For ( x , y , z ) ∈ F i , we note that one of the two equalities( x j , x k ) = ( y j , − y k ) or ( x j , x k ) = ( z k , − z j ) holds in P . Hence p : X → Y restricts to an isomor-phism from F i to Y .The α -invariant is defined by means of Cartier divisors. As X is smooth, we may also view suchdivisors as Weil divisors [13, p. 141] and regard them as members of the free abelian group Div X generated by the prime divisors. We may then extend this group to the group Div R X = Div X ⊗ Z R of R -divisors and consider the submonoid of effective R -divisors (see [15, p. 48]). The pseudoeffectivecone C eff ( X ) ⊂ Pic X ⊗ R is the closure of the convex cone spanned by the classes of all effective R -divisors on X (see [15, p. 47]). The main result of this subsection is Proposition 7.1 below, assertingthat C eff ( X ) is spanned by the nine classes [ D i ] , [ E i ] , [ F i ], 1 ≤ i ≤ 3. We start with the followinglemma. Lemma 7.1. The group G m acts trivially on Pic X .Proof. The surface Y is a del Pezzo surface of degree six and Pic Y is spanned by the classes of itssix lines. The image p ∗ (Pic Y ) of the functorial map p ∗ : Pic Y → Pic X is therefore spanned by theclasses of all D i and E i . As D i and E i are G m -invariant, G m acts trivially on p ∗ (Pic Y ).Next, let L = pr ∗ ( O P (1)) be the sheaf associated to the first projection pr of X ⊂ P × P × P .Then, [ L ] + p ∗ (Pic Y ) generates Pic X/p ∗ (Pic Y ) ∼ = Z as X is a P -bundle over Y . As G m actstrivially on [ L ] and p ∗ (Pic Y ), it acts trivially on Pic X . (cid:3) The following lemma will make it easier to determine C eff ( X ). Lemma 7.2. An effective divisor on X is linearly equivalent to a G m -invariant effective divisor on X . Proof. An effective divisor D on X is given by the vanishing of a global section of the invertible O X -module L = O X ( D ). As G m stabilizes the class of L in Pic X , it therefore follows from theproof of [17, Prop. 1.5, p. 34] that we may endow L with a G m -linearization (see also [13, Prop.2.3]). This is equivalent to a lifting of the G m -action on X to a G m -action on the line bundle L → X defined by L (see [17, p. 31]). There is thus an induced rational representation of G m on H ( X, L ).Since G m is diagonalizable ([25, p. 21]), this induced rational representation must be a direct sum ofone-dimensional ones. Hence there is a G m -invariant one-dimensional subspace S of H ( X, L ). Thedivisor of zeros of S ([13, p. 157]) is then a G m -invariant effective divisor on X linearly equivalentto D , as desired. (cid:3) In the following we will use G m -linearizations on invertible sheaves L on P × P and Z compatiblewith (7.1). To construct such a linearization on L = O P × P ( m, n ) , let h m, n i = (cid:0) m +33 (cid:1)(cid:0) n +33 (cid:1) − h m,n : P × P → P h m,n i be the morphism defined by all monomials of bidegree ( m, n ) in ( x , x , x ; z , z , z ). Then we have O P × P ( m, n ) = h ∗ m,n O P h m,n i (1), and there is a natural G m -action on H ( P h m,n i , O P h m,n i (1)) givenby(7.3) ( γ G )( x , z ) = G ( γ x , γ x , γ x ; γ − z , γ − z , γ − z )for homogeneous polynomials G ( x , z ) of bidegree ( m, n ). This G m -action gives rise to a G m -linearization on O P (1), which may be pulled back to a G m -linearization on O P × P ( m, n ) = h ∗ m,n O P h m,n i (1)(see [17, Prop. 1.7, p. 34]). Similarly, by considering the restriction of h m,n to Z , we obtain a G m -linearization on O Z ( m, n ) such that the induced restriction from H ( P × P , O P × P ( m, n )) to H ( Z, O Z ( m, n )) is G m -equivariant. Lemma 7.3. Let ∆ be a G m -invariant effective divisor on Z . Then there exists a one-dimensional G m -invariant subspace S of H ( P × P , O P × P ( m, n )) such that ∆ is the divisor of the section s Z ∈ H ( Z, O Z ( m, n )) for any s ∈ S \ { } .Proof. Every effective divisor on Z is the divisor div( σ ) of some global section σ of an invertiblesheaf L on Z . It is well known (cf. e.g. [23, Th. 2.4]) that any invertible sheaf on Z is isomorphic tosome O Z ( m, n ) where m, n ≥ H ( Z, O Z ( m, n )) = 0. We may and shall thus assume that L = O Z ( m, n ) for m, n ≥ 0. An effective divisor ∆ on Z will then correspond to a one-dimensionalsubspace Σ of H ( Z, O Z ( m, n )) for some m, n ≥ G m -invariant ifand only if Σ is G m -invariant.We now apply the K¨unneth formula in [22] to pr ∗ ( O P ( k )) ⊗ pr ∗ ( O P ( l )) . We then obtain that H ( P × P , O P × P ( k, l )) = 0 as H ( P , O P ( k )) = 0 for all k . In particular, we conclude that H ( P × P , O P × P ( m − , n − H ( P × P , O P × P ( m, n ))to H ( Z, O Z ( m, n )) must be surjective.As this restriction map is G m -equivariant and the G m -representation on H ( P × P , O P × P ( m, n ))is a direct sum of one-dimensional ones, there is some one-dimensional G m -invariant subspace of H ( P × P , O P × P ( m, n )), which restricts to Σ on Z . (cid:3) We prepare for the statement of the next lemma with a definition. A bihomogeneous polyno-mial G ( x , z ) of bidegree ( m, n ) is said to be an eigenpolynomial under G m if it is contained ina one-dimensional G m -invariant subspace of H ( P × P , O P × P ( m, n )). In other words, G is aneigenpolynomial if and only if for each γ ∈ G m , we can find a constant c such that γ G = cG underthe action in (7.3). In the following lemma, monomials in three variables occur. We write these inthe compact form x a = x a x a x a for a ∈ N . Confusion with the notation (1.13) that was used inthe analytic part should not arise. Lemma 7.4. Let G ( x , z ) be a bihomogeneous eigenpolynomial under G m . Then there exists a mono-mial M = x e z f and a ternary homogeneous polynomial H such that G = M H ( x z , x z , x z ) . HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 33 Proof. Let I be the set of all sixtuples ( a , a , a , b , b , b ) such that x a z b is a monomial in G with non-zero coefficient. As γ M = γ a − b M for M = x a z b , the characters sending γ ∈ G m to γ a − b ∈ G m coincide for all sixtuples in I . The triples ( a − b , a − b , a − b ) and the sixtuples( e , e , e , f , f , f ) with e i = max( a i − b i , , f i = max( b i − a i , I . Hence, defining M = x e z f , we get that M = M Q i ( x i z i ) min( a i ,b i ) for any monomial M = x a z b in G with non-zero coefficient. Thus there existsa homogeneous polynomial H of degree X i =1 min( a i , b i ) = 12 X i =1 ( a i + b i − e i − f i ) ≥ G = M H ( x z , x z , x z ). (cid:3) We now consider the images g ∗ (∆) ∈ Div X of effective divisors ∆ ∈ Div Z under the functorialmap g ∗ : Div Z → Div X . Lemma 7.5. Let H ( t , t , t ) be a ternary homogeneous polynomial of degree n not divisible by t + t + t , and let ∆ ∈ Div Z be the effective divisor defined by H ( x z , x z , x z ) . Then themultiplicity of D i in g ∗ (∆) ∈ Div X is equal to n for i = 1 , and .Proof. Let Z ⊂ Z be the subscheme associated to ∆ (cf. [13, p. 145]) and let l , l , l be the threelines on Z described in (7.2). Then, D + D + D is the exceptional divisor (cf. [11, App. B6]) ofthe blow-up g : X → Z at l ∪ l ∪ l . Therefore, the multiplicity of D i in g ∗ (∆) must be equal tothe multiplicity m i of Z along l i (see [11, p. 79] for the definition of m i and [13, Ch. 5, Prop. 5.3]for the proof of a similar statement).It suffices to prove the assertion for D and we may also use the equation x z + x z + x z = 0 for Z to eliminate t = − t − t . This replaces H ( t , t , t ) by a non-zero binary form G ( t , t ). Then, Z is the subscheme of P × P defined by ( x z + x z + x z , G ( x z , x z )) and l the subschemedefined by ( z , z , x ). It is now clear from the definition of m that m = n , as G ( x z , x z ) is ofdegree n with respect to ( z , z ). (cid:3) We are now in a position to determine C eff ( X ). Recall that Pic X is a free abelian group of rankfive ([4, Theorem 4]). Proposition 7.1. The pseudoeffective cone C eff ( X ) is spanned by the nine classes [ D i ] , [ E i ] , [ F i ] , ≤ i ≤ .Proof. By Lemma 7.2, it is enough to show that the class [ D ] ∈ Pic X of any G m -invariant effectivedivisor D on X is in the cone spanned by the nine classes above. To do this, it suffices to treatthe case where none of D , D , D occur in the prime decomposition of D as D , D and D are G m -invariant.Now let D U be the restriction of D to U = X \ ∪ i =1 D i and let g U : U → Z be the restrictionof g to U . Then g U is an open immersion with Z \ g U ( U ) of codimension two in Z . The functorialmap g ∗ U : Div Z → Div U is thus an isomorphism, which restricts to an isomorphism betweenthe submonoids of G m -invariant effective divisors on Z and U . There are therefore a unique G m -invariant effective divisor ∆ on Z with g ∗ U (∆) = D U and unique non-negative integers n i with g ∗ (∆) = D + P i =1 n i D i .By Lemma 7.3 and Lemma 7.4 there is a decomposition ∆ = ∆ ′ + ∆ ′′ into two G m -invarianteffective divisors on Z where ∆ ′ is defined by a monomial x e y f and ∆ ′′ by H ( x z , x z , x z ) for aternary form H ( t , t , t ). As the divisors of x i (resp. z i ) are given by D i + F i (resp. D j + D k + E i ),we infer that g ∗ (∆ ′ ) = X i =1 e i ( D i + F i ) + X i =1 f i ( D j + D k + E i ) . By Lemma 7.5 we have also a decomposition g ∗ (∆ ′′ ) = n ( D + D + D ) + D ∗ where n = deg H and D ∗ is an effective divisor where D , D and D do not occur. By adding these two decompositionsand comparing the result with g ∗ (∆) = D + P i =1 n i D i , we obtain that D = D ∗ + X i =1 e i F i + X i =1 f i E i . Moreover, as g ∗ (∆ ′′ ) is linearly equivalent to the divisor n ( D i + F i ) + n ( D j + D k + E i ) of x ni z ni ,we get that [ D ∗ ] = n [ E i + F i ] for any i ∈ { , , } and that [ D ] belongs to the cone spanned by[ E ] , [ E ] , [ E ] , [ F ] , [ F ] and [ F ]. (cid:3) Computation of α ( X ) . In this section we compute Peyre’s α -invariant (see [18, Def. 2.4]) for X . To do this, we let D (resp. D ) be the effective divisors given by L ( z ) = 0 (resp. M ( x ) = 0) fortwo fixed ternary linear forms L and M . We then have the following linear equivalences(7.4) E i ∼ D − D j − D k as div( z i = 0) ∼ D , and(7.5) F i ∼ D − D i as div( x i = 0) ∼ D . Lemma 7.6. The divisor D − D − D − D + 2 D is an anticanonical divisor on X .Proof. The canonical sheaf ω V is isomorphic to O V ( − , − 1) as V is of bidegree (1 , ω X = f ∗ ω for the morphism f : X → V . Hence the divisor 2 D + ( D i + E j + E k ) of M ( x ) y i is anticanonical. Moreover, 2 D + D i + E j + E k ∼ D − D − D − D + 2 D by (7.4), thereby completing the proof. (cid:3) Now let C eff ( X ) ∨ ⊂ Hom(Pic X ⊗ R , R ) be the dual cone of all linear maps Λ : Pic X ⊗ R → R suchthat Λ([ D ]) ≥ D on X , and let l : Hom(Pic X ⊗ R , R ) → R be the linearmap which sends Λ to Λ([ − K X ]). We then endow Hom(Pic X ⊗ R , R ) with the Lebesque measured s normalized such that the lattice Hom(Pic X, Z ) has covolume 1, and we endow H X = l − (1)with the measure d s/ d l . Explicitly, if w , . . . , w are coordinates for Hom(Pic X ⊗ R , R ) = R withrespect to a Z -basis of L and l ( w , . . . , w ) = α w + . . . + α w , thend s d l = d w . . . d d w i . . . d w / | α i | whenever α i = 0.After these preparations, we can now define α ( X ) as α ( X ) = Z C eff ( X ) ∨ ∩ H X d s d l . If we let e , . . . , e be the Z -basis of L with e i ([ D j ]) = δ ij , then we have the following Lemma 7.7. (a) The hyperplane H X ⊂ R is defined by the equation w − w − w − w + 2 w = 1 . (b) The dual cone C eff ( X ) ∨ is defined by the inequalities w ≥ w i ≥ , ≤ i ≤ w − w i − w j ≥ , ≤ i < j ≤ . Proof. (a) One has l ( w , w , w , w , w ) = P i =0 e i ([ − K X ]) w j by the definition of l . Hence l =2 w − w − w − w + 2 w by Lemma 7.6.(b) One has P i =0 w i e i ∈ C eff ( X ) ∨ if and only if P i =0 w i e i ([ D ]) ≥ D ] ∈ C eff ( X ). Hence,by Proposition 7.1 we have that ( w , w , w , w , w ) ∈ C eff ( X ) ∨ if and only if P i =0 w i e i ([ D ]) ≥ HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 35 for any D ∈ { D , D , D , E , E , E , F , F , F } . Now by using (7.4) and (7.5) and e i ([ D j ]) = δ ij ,we conclude that these nine inequalities are the same as the inequalities in the statement of thelemma. (cid:3) We are now prepared to compute the α -invariant of X : Proposition 7.2. One has α ( X ) = 12 = 1576 . Proof. Eliminating w = (1 + w + w + w − w ) / w , w , w ,we obtain from Lemma 7.7 that α ( X ) = 12 · · vol( Q ) , where Q ⊂ R is defined by the inequalities w ≥ w ≥ w ≥ w ≥ w + w + 2 w ≤ w . Changing variables by the unimodular linear transformation v = w − w , v = w − w , v = w , v = w − w , we find that α ( X ) = 3vol( P ), where P ⊂ [0 , ∞ ) is defined by 3 v + 4 v + 3 v + 2 v ≤ 1. Hence, α ( X ) = 14! 33 · · · . (cid:3) The adelic volume of X We keep the notation of the previous chapter. The aim of this chapter is to give an explicitdescription of Peyre’s Tamagawa measure µ H on X ( A ) = X ( R ) × Q p X ( Q p ), and to compute thevolume µ H ( X ( A )). The interest in this comes from Peyre’s prediction [19] that the constant c inthe expected asymptotic formula N ( B ) = cB (log B ) rk Pic X − (1 + o (1))should be given by c = α ( X ) µ H ( X ( A )).8.1. Heights and adelic metrics. The morphism f : X → V restricts to an isomorphism betweenthe open subsets X ◦ ⊂ X and V ◦ ⊂ V defined by x x x y y y = 0. We conclude that N ( B ) = |{ w ∈ V ◦ ( Q ) : H ( w ) ≤ B }| = |{ x ∈ X ◦ ( Q ) : H ( f ( x )) ≤ B }| where the height H : V ( Q ) → N was defined in (1.2) for a certain choice of representatives and canalso be written as H ( x , y ) = Y v max ≤ i,j ≤ (cid:12)(cid:12) x i y j (cid:12)(cid:12) v . The aim of this section is to reinterpret this height and H ◦ f : X ( Q ) → N in terms of adelicmetrics on ω − V and ω − X as in [18]. These metrics will be constructed by means of global sectionson ω − V and ω − X = f ∗ ( ω − V ), which we obtain by glueing local sections on the open subsets V i,j ⊂ V and X i,j ⊂ X where x i y j = 0.We write ( P × P ) i,j for the open subset of P × P where x i y j = 0. On this set, we shall useaffine coordinates. For k = i and l = j these are given by x ( i ) k = x k x i and y ( j ) l = y l y j . Then V i,j ⊂ ( P × P ) i,j is the affine hypersurface in A defined by F ij = 0, where F i,j ( x ( i ) i +1 , x ( i ) i +2 , y ( j ) j +1 , y ( j ) j +2 ) = x ( i )1 y ( j )2 y ( j )3 + x ( i )2 y ( j )1 y ( j )3 + x ( i )3 y ( j )1 y ( j )2 ; here and in the following we put x ( i ) i = y ( j ) j = 1 and we interpret indices i, j, k in Z / Z .There is a unique global section s of ω P × P ( D ) which for any choice of i, j restricts to s ( P × P ) i,j = d x ( i ) i +1 x ( i ) i +1 ∧ d x ( i ) i +2 x ( i ) i +2 ∧ d y ( j ) j +1 y ( j ) j +1 ∧ d y ( j ) j +2 y ( j ) j +2 ∈ Γ (cid:0) ( P × P ) i,j , ω P × P ( D ) (cid:1) . This can be seen directly because one hasd x ( i ) i +1 x ( i ) i +1 ∧ d x ( i ) i +2 x ( i ) i +2 = d x ( k ) k +1 x ( k ) k +1 ∧ d x ( k ) k +2 x ( k ) k +2 on the open subset of P where x i x k = 0. Alternatively, this claim is a special case of a generalresult for toric varieties (see [4, Lemma 12]). The latter result also shows that s is a global generatorof the O P × P -module ω P × P ( D ).Now put F = x y y + x y y + x y y , and then define(8.1) ω i,j = x x x y y y x i y j F s ∈ Γ (cid:0) ( P × P ) , ω P × P ( V + 2 H x i + H y i ) (cid:1) , where H x i (resp. H y i ) are the prime divisors on P × P defined by the vanishing of x i (resp. y j ).Then, ω i,j is a global generator of ω P × P ( V + 2 H x i + H y i ) with ω i,j = 1 F i,j d x ( i ) i +1 ∧ d x ( i ) i +2 ∧ d y ( j ) j +1 ∧ d y ( j ) j +2 on ( P × P ) i,j .We now consider the Poincar´e residue map Res: ω P × P ( V ) → ι ∗ ω V for the inclusion map ι : V → P × P . The Poincar´e residue map is usually given as a homomorphism Ω nW ( V ) → ι ∗ Ω n − V for theinclusion map ι : V → W of a non-singular hypersurface V ⊂ W in an n -dimensional non-singularvariety (cf. [Re3, p. 89], for example). More generally, one can also use Poincar´e residues to definelocal sections on the canonical sheaf ω V of an arbitrary normal hypersurface V ⊂ W (cf. [We]) asone still gets regular ( n − V ns of V and since ω V = j ∗ Ω n − V ns forthe inclusion map j : V ns → V . After these general remarks we return to our specific situation.By regarding ω i,j as a local section of ω P × P ( V ) on ( P × P ) i,j we obtain a local sectionRes( ω i,j ) ∈ Γ( V i,j , ω V ), whereRes( ω i,j ) = ( − pos( z )+1 ∂F i,j /∂z d x ( i ) i +1 ∧ . . . c d z . . . ∧ d y ( j ) j +2 , on the open subset of V i,j where ∂F i,j /∂z = 0 and pos( z ) ∈ { , , , } is the position of z ∈{ x ( i ) i +1 , x ( i ) i +2 , y ( j ) j +1 , y ( j ) j +2 } . This defines Res( ω i,j ) on the non-singular locus U i,j of V i,j with Res( ω i,j ) =0 everywhere on U i,j . As V i,j is normal, we may then extend Res( ω i,j ) to a volume form on V i,j bya standard argument (see [13, p. 181]).Hence there is an inverse nowhere vanishing local section τ i,j = Res( ω i,j ) − ∈ Γ( V i,j , ω − V ) with τ i,j = ( − pos( z )+1 ∂F i,j /∂z ∂∂x ( i ) i +1 ∧ . . . c ∂∂z . . . ∧ ∂∂y ( j ) j +2 on the non-singular locus of V i,j .We shall also write σ i,j ∈ Γ( X i,j , ω − X ) for the local section corresponding to f ∗ τ i,j := τ i,j ⊗ ∈ Γ (cid:0) f − ( V i,j ) , f − ω − V ⊗ f − O V O X (cid:1) = Γ (cid:0) X i,j , f ∗ ω − V (cid:1) . As τ i,j ∈ Γ( V i,j , ω − X ) is inverse to the volume form Res( ω i,j ) on V i,j , we conclude that σ i,j is inverseto the volume form σ − i,j on X i,j corresponding to f ∗ (Res( ω i,j )) = Res( ω i,j ) ⊗ ∈ Γ (cid:0) f − ( V i,j ) , f − ω V ⊗ f − O V O X (cid:1) = Γ ( X i,j , f ∗ ω V ) . HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 37 Lemma 8.1. Let i, j, k, l ∈ { , , } . (a) We have τ i,j = (cid:0) x ( k ) i (cid:1) y ( l ) j τ k,l on V i,j ∩ V k,l . (b) We have σ i,j = (cid:0) x ( k ) i (cid:1) y ( l ) j σ k,l on X i,j ∩ X k,l .Proof. (a) By (8.1) we have ω i,j = (cid:0) x ( i ) k (cid:1) y ( j ) l ω k,l on V i,j ∩ V k,l . Hence Res( ω i,j ) = (cid:0) x ( i ) k (cid:1) y ( j ) l Res( ω k,l )and τ i,j = (cid:0) x ( k ) i (cid:1) y ( l ) j τ k,l on V i,j ∩ V k,l .(b) Let a ∈ Γ( X i,j ∩ X k,l , O X ) be the image of a ∈ Γ( X i,j ∩ X k,l , O X ) = Γ( X i,j ∩ X k,l , f − O V ) underthe natural map from f − O V to O X . Then, f ∗ ( aτ k,l ) = aτ k,l ⊗ a ( τ k,l ⊗ 1) = af ∗ ( τ k,l )on X i,j ∩ X k,l . Hence f ∗ ( τ i,j ) = (cid:0) x ( k ) i (cid:1) y ( l ) j f ∗ ( τ k,l ) on X i,j ∩ X k,l by (a), thereby proving the assertionin (b). (cid:3) The lemma implies that τ i,j ∈ Γ( V i,j , ω − V ) extends to a global anticanonical section that we stilldenote by τ i,j ∈ Γ( V, ω − V ). Similarly, we let σ i,j be the global anticanonical section on X definedby σ i,j = f ∗ τ i,j .For each place v of Q , the global sections τ i,j , 1 ≤ i, j ≤ 3, define a v -adic norm on ω − V with(8.2) k τ ( w v ) k v = min i,j (cid:12)(cid:12)(cid:12)(cid:12) ττ i,j ( w v ) (cid:12)(cid:12)(cid:12)(cid:12) v = min i,j | τ Res( ω i,j ) | v for a local section τ of ω − V defined at w v ∈ V ( Q v ). Here the minimum is taken over all i, j ∈ { , , } such that τ i,j ( w v ) = 0. This definition is similar to the definition in [18, pp. 107-108], although it iscalled a v -adic metric there.In the same way we may define a v -adic norm on ω − X by letting(8.3) k σ ( x v ) k v = min i,j (cid:12)(cid:12)(cid:12)(cid:12) σσ i,j ( x v ) (cid:12)(cid:12)(cid:12)(cid:12) v . for a local section σ of ω − X defined at x v ∈ X ( Q v ). Here now the minimum is taken over all i, j ∈ { , , } such that σ i,j ( x v ) = 0. We then have, just as in [4, Lemma 15], the following result. Lemma 8.2. (a) Let w ∈ V ( Q ) and τ be a local section of ω − V with τ ( w ) = 0 . Then H ( w ) = Q v k τ ( w ) k − v . (b) Let x ∈ X ( Q ) and σ be a local section of ω − X with σ ( x ) = 0 . Then H ( f ( x )) = Q v k σ ( x ) k − v .Proof. On applying the product formula Q v | α | v = 1 for α ∈ Q ∗ , it suffices in both cases to prove theformula for one local section. To prove (a), suppose that w ∈ V k,l and let τ = τ k,l . Then τ ( w ) = 0,and by (8.2) and Lemma 8.1(a) we see that k τ ( w ) k − v = max i,j (cid:12)(cid:12)(cid:12)(cid:12) τ i,j τ k,l ( w v ) (cid:12)(cid:12)(cid:12)(cid:12) v = 1 | x k y l | v max i,j (cid:12)(cid:12) x i y j (cid:12)(cid:12) v holds for each place v . Hence the desired identity Q v k τ ( w ) k − v = H ( w ) follows from the productformula.To prove (b), we may assume that x ∈ X k,l and choose σ = σ k,l . The proof is then the same asfor (a), but based on using (8.3) and Lemma 8.1(b). (cid:3) The volume of the adelic space X ( A ) . We now describe Peyre’s Tamagawa measure µ H on X ( A ) = X ( R ) × Q p X ( Q p ) defined by the adelic metric on ω − X of all v -adic norms in (8.3), andcompute the volume of the adelic space X ( A ) with respect to this measure.To obtain this measure, we recall the definition in [18, (2.2.1)] of a measure µ v on X ( Q v ) associatedto a v -adic norm on ω − X . Let | σ − i,j | v be the v -adic density on X i,j ( Q v ) of the volume form σ − i,j on X i,j . Then, for a Borel subset N v of X i,j ( Q v ), and with the v -adic norm on ω − X defined in (8.3),we put µ v ( N v ) = Z N v | σ − i,j | v max k,l (cid:12)(cid:12) σ k,l σ − i,j (cid:12)(cid:12) v . This defines a measure µ v on X ( Q v ). On applying Lemma 8.1(b), we may rewrite this as µ v ( N v ) = Z N v | σ − i,j | v max k,l (cid:12)(cid:12)(cid:0) x ( i ) k (cid:1) y ( j ) l (cid:12)(cid:12) v . As usual, we write µ ∞ = µ v when Q v = R and µ p = µ v when Q v = Q p . Lemma 8.3. Let D = n w ∈ V ◦ ( R ) : | x | ≤ | x | , | x | ≤ | x | , | y | ≥ | y | , x (3)1 > , y (3)1 > o . Then µ ∞ ( X ( R )) = 24 Z D (cid:12)(cid:12) Res (cid:0) d x (3)1 ∧ d x (3)2 ∧ d y (3)1 ∧ d y (3)2 (cid:1)(cid:12)(cid:12) max (cid:0) y (3)1 , (cid:1) . Proof. As the inverse image f ∗ (cid:0) τ − i,j (cid:1) of the volume form τ − i,j = Res( ω i,j ) on V i,j is sent to thevolume form σ − i,j on X i,j under the canonical map from f ∗ ω V → ω X , we have thus for Borel subsets N ⊂ X ◦ ( R ) that(8.4) µ ∞ ( N ) = Z f ( N ) | Res( ω i,j ) | max k,l | σ k,l Res( ω i,j ) | = Z f ( N ) | Res( ω i,j ) | max k,l (cid:12)(cid:12)(cid:0) x ( i ) k (cid:1) y ( j ) l (cid:12)(cid:12) . for any fixed i, j ∈ { , , } .The hyperoctahedral group Z ≀ S of order 2 × 3! acts on the affine hypersurface in A defined by F = 0. This group consists of signed symmetries over ̺ ∈ S sending ( x i , y i ) to one of ( x ̺ ( i ) , y ̺ ( i ) )or − ( x ̺ ( i ) , y ̺ ( i ) ) for i ∈ { , , } and we obtain in this way an action of Z ≀ S on V as well. Asthe symmetry sending all ( x , y ) to − ( x , y ) is trivial on V , we get in fact a faithful action of theoctahedral group O of order 24 on V , which preserves V ◦ . The set D is a fundamental domain forthe (measure-preserving) action of this group, hence µ ∞ ( V ◦ ( R )) = 24 µ ∞ ( D ).We now apply (8.4) with i = j = 3. Then ω , = d x (3)1 ∧ d x (3)2 ∧ d y (3)1 ∧ d y (3)2 andmax ≤ k ≤ (cid:12)(cid:12)(cid:0) x (3) k (cid:1) (cid:12)(cid:12) max ≤ l ≤ (cid:12)(cid:12) y (3) l (cid:12)(cid:12) = max (cid:16)(cid:12)(cid:12) y (3)1 (cid:12)(cid:12) , (cid:17) on D . Hence µ ∞ ( D ) = Z D (cid:12)(cid:12) Res (cid:0) d x (3)1 ∧ d x (3)2 ∧ d y (3)1 ∧ d y (3)2 (cid:1)(cid:12)(cid:12) max (cid:0)(cid:12)(cid:12) y (3)1 (cid:12)(cid:12) , (cid:1) and we are done. (cid:3) We are now prepared to compute µ ∞ ( X ( R )) explicitly. This is the counterpart to Lemma 2.10. Lemma 8.4. We have µ ∞ ( X ( R )) = 96 log 2 − 12 + 4 π .Proof. Set t = x (3)1 , t = x (3)2 , u = y (3)1 and u = y (3)2 . Then F , = t u + t u + u u and | Res( ω , ) | = d t d u d t | ∂F , /∂u | = d t d u d t | t + u | . Moreover, we have the equivalences | y | ≥ | y | ⇐⇒ | u | ≥ | u | ⇐⇒ | t | ≤ | t + u | HE MANIN-PEYRE FORMULA FOR A CERTAIN BIPROJECTIVE THREEFOLD 39 as − u t = ( t + u ) u on V . By the previous lemma we conclude that µ ∞ ( X ( R ))24 = Z ∞ Z − Z min(1 , | t + u | ) t =0 d t | t + u | d t d u max ( u , Z ∞ Z − min (cid:18) | t + u | , (cid:19) d t d u max ( u, Z − u + log( u + 1)max ( u, 1) d u + Z ∞ log (cid:18) u + 1 u − (cid:19) d uu , and a straightforward computation now shows that this quantity equals 4 log 2 − + π , as desired. (cid:3) To compute the p -adic volumes µ p ( X ( Q p )), we shall make use of the scheme X ⊂ P Z × P Z × P Z with coordinates ( x , y , z ) defined by the equations x z + x z + x z = 0 and y z = y z = y z .It is smooth over Z , and there is an extension of f : X → V to a morphism f : X → V with f ( x , y , z ) = ( x , y ) onto the subscheme V ⊂ P Z × P Z defined by x y y + x y y + x y y = 0.There is also a functorial homomorphism f ∗ ω V / Z → ω X/ Z of invertible O X -modules for the relativecanonical (or dualising) invertible sheaves ω V / Z and ω X/ Z , which must be an isomorphism as f andthe base extensions f F p : X F p → V F p are crepant (see [4, Theorem 4]).Now consider the dual isomorphism from ω − X/ Z to f ∗ ω − V / Z . It extends the isomorphism from ω − X to f ∗ ω − V that was used to define σ i,j = f ∗ τ i,j . We may thus extend σ i,j to global sections σ i,j = f ∗ τ i,j of ω − X/ Z = f ∗ ω − V / Z as follows. We first define τ i,j and σ i,j on the principal open subsetswhere x i y j = 0 in the same way as we defined σ i,j and τ i,j , and we then use an analogue of Lemma8.1 for V and X to extend these sections to global sections. Lemma 8.5. For all primes p one has µ p ( X ( Q p )) = 1 + 5 p + 5 p + 1 p . Proof. In [21, Def. 2.9] there is defined a measure m p on X ( Z p ) called the model measure forwhich m p ( X ( Z p )) = | X ( F p ) | /p dim X (see [21, Cor. 2.15]). As X × Z F p is a P -bundle over a delPezzo surface B of degree 6 over F p , we conclude that | X ( F p ) | = (cid:0) p + 4 p + 1 (cid:1) ( p + 1) and m p ( X ( Z p )) = 1 + 5 p + 5 p + 1 p . As X is proper over Z , there is a natural bijection X ( Z p ) = X ( Q p ). To complete the proof of thelemma, it is thus enough to show that m p = µ p . The definitions of m p and µ p are both based onPeyre’s construction [18, (2.2.1)] of a measure on X ( Q v ) associated to a v -adic norm on ω − X . For m p one uses a p -adic norm k · k ∗ p called the model norm, as described in [21, (2.9)]. Thus it onlyremains to prove that this norm coincides with the p -adic norm k · k p in (8.3) used to define µ p .Therefore, let σ be a local section of ω − X defined at x p ∈ X ( Q p ). To show that k σ k ∗ p = k σ k p in aneighbourhood of x p , we choose i, j such that x p ∈ X i,j ( Q p ). The restriction of ω − X to U = X i,j isa free O U -module generated by σ i,j as σ i,j is the inverse to the volume form f ∗ (Res( ω i,j )) on X i,j .By the same argument one obtains that the restriction of ω − X/ Z to U = X i,j is a free O U -modulegenerated by σ i,j . As σ i,j restricts to σ i,j on X i,j , we conclude from the definition of the modelnorm (see [21, 1.9 and 2.9]) that k σ i,j k ∗ p = 1 on X i,j , and by (8.3) that | σ i,j | p = 1 on X i,j . Hence k σ k ∗ p = | σ/σ i,j | p k σ i,j k ∗ p = | σ/σ i,j | p k σ i,j k p = k σ k p in a neighbourhood of x p , as was to be shown.Now let L p ( s, Pic X ) = det(1 − p − s Fr p | Pic (cid:16) X F p ) ⊗ Q (cid:17) − . Then, as Gal( F p / F p ) acts trivially on Pic ( X F p ) ∼ = Z , we conclude that L p ( s, Pic X ) = (1 − p − s ) − , so that L ( s, Pic X ) = Y all p L p ( s, Pic X ) = ζ ( s ) In particular, lim s → ( s − L ( s, Pic X ) = 1 and L p (1 , Pic X ) − = (cid:18) p − p (cid:19) . Peyre’s measure µ H on X ( A ) = X ( R ) × Q p X ( Q p ) is therefore given by µ H = µ ∞ × Y p (cid:16) p − p (cid:17) µ p , and it is shown in [19, Def. 4.6] that this gives a well defined measure on X ( A ). As X ( Q ) is densein X ( A ), we conclude (see [19, Def. 4.8]) that(8.5) µ H ( X ( A )) = µ ∞ ( X ( R )) Y p (cid:18) p − p (cid:19) µ p ( X ( Q p )) . We may now combine the conclusions of Lemma 8.4, Lemma 8.5 and (8.5) to infer the followingresult. Proposition 8.1. Peyre’s Tamagawa constant τ H ( X ) = µ H ( X ( A )) associated to the adelic metricof all v -adic norms in (8.3) is given by τ H ( X ) = (cid:0) 96 log 2 − 12 + 4 π (cid:1) Y p (cid:18) p + 5 p + 1 p (cid:19) . The leading term of the asymptotic formula. We finally show that the asymptotic formulafor N ( B ) in Theorem 1.1 is in accordance with conjectures made in [19]. As the biprojective threefold V defined by (1.1) is singular, we cannot refer to the original conjectures of Manin [10] and Peyre[18]. To overcome this, we make use of the observation in Section 8.1 that N ( B ) = { x ∈ X ◦ ( Q ) : ( H ◦ f )( x ) ≤ B } . We have also seen in (8.3) that H ( f ( x )) = Y v k σ ( x ) k − v for a local anticanonical section σ with σ ( x ) = 0. We may therefore refer to the conjectures ofPeyre [19] for “almost” Fano varieties instead. The following result shows that X satisfies the threeconditions for being such a variety. Lemma 8.6. Let X ⊂ P × P × P be as before. Then (a) H ( X, O X ) = H ( X, O X ) = 0 ; (b) The geometric Picard group Pic ( X Q ) is torsion-free; (c) The anticanonical class is in the interior of C eff ( X ) .Proof. To prove (a), we apply the Leray spectral sequence H i ( Y, R j p ∗ O X ) = ⇒ H i + j ( X, O X ) tothe P -bundle p : X → Y . Then, we obtain isomorphisms H i ( Y, O Y ) = H i ( X, O X ) for all i , with H ( Y, O Y ) = H ( Y, O Y ) = 0 for a del Pezzo surface.For (b), we use that X is a P -bundle over a del Pezzo surface Y of degree 6. This givesPic( X Q ) ∼ = Pic( Y Q ) ⊕ Z ∼ = Z .Finally (c) follows from Lemma 7.7 and the fact that 3 D i + E j + E k + F i is an anticanonicaldivisor if { i, j, k } = { , , } (see Lemma 7.6). 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Titchmarsh, The theory of the Riemann zeta-function , Oxford University Press, 1986 Mathematisches Institut, Bunsenstr. 3-5, 37073 G¨ottingen, Germany E-mail address : [email protected] E-mail address : [email protected] Mathematical Sciences, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden E-mail address ::